Trigonometric / Hyperbolic Identities

1. Trigonometric Identities for Unit Circle
   1. \({\sin}^2 \theta + {\cos}^2 \theta=1\hspace{.3cm}\Rightarrow {\cos}^2 \theta=1-{\sin}^2 \theta\hspace{.3cm}\Rightarrow {\sin}^2 \theta=1-{\cos}^2 \theta\)
   2. \({\sec}^2 \theta - {\tan}^2 \theta=1\hspace{.3cm}\Rightarrow {\sec}^2 \theta=1+{\tan}^2 \theta\hspace{.3cm}\Rightarrow {\tan}^2 \theta=1+{\sec}^2 \theta\)
   3. \({\csc}^2 \theta - {\cot}^2 \theta=1\hspace{.3cm}\Rightarrow {\csc}^2 \theta=1+{\cot}^2 \theta\hspace{.3cm}\Rightarrow {\cot}^2 \theta=1+{\csc}^2 \theta\)
2. Hyperbolic Identities for Unit Hyperbola
   1. \({\cosh}^2 \theta - {\sinh}^2 \theta=1\hspace{.3cm}\Rightarrow {\cosh}^2 \theta=1+{\sinh}^2 \theta\hspace{.3cm}\Rightarrow {\sinh}^2 \theta={\cosh}^2 \theta-1\)
   2. \({\text{sech}}^2 \theta + {\tanh}^2 \theta=1\hspace{.3cm}\Rightarrow {\text{sech}}^2 \theta=1-{\tanh}^2 \theta\hspace{.3cm}\Rightarrow {\tanh}^2 \theta=1-{\text{sech}}^2 \theta\)
   3. \({\coth}^2 \theta-{\text{csch}}^2 \theta =1\hspace{.3cm}\Rightarrow {\text{csch}}^2 \theta={\coth}^2\theta -1 \hspace{.3cm}\Rightarrow {\coth}^2 \theta=1+{\text{csch}}^2 \theta\)
3. Since \(e^{i\theta}=\cos\theta + i \sin\theta\) and \(e^{-i\theta}=\cos\theta - i \sin\theta\) therefore
   
   \(\cos\theta= \frac{e^{i\theta} + e^{-i\theta}}{2}\hspace{.5cm}\sin\theta= \frac{e^{i\theta} - e^{-i\theta}}{2i}\hspace{.5cm}\tan\theta= \frac{e^{i\theta} - e^{-i\theta}}{i(e^{i\theta} + e^{-i\theta})}\)
   
   
4. Since \(e^{\theta}=\cosh\theta + \sinh\theta\) and \(e^{-\theta}=\cosh\theta - \sinh\theta\) therefore
   
   \(\cosh\theta= \frac{e^{\theta} + e^{-\theta}}{2}\hspace{.5cm}\sinh\theta= \frac{e^{\theta} - e^{-\theta}}{2}\hspace{.5cm}\tanh\theta= \frac{e^{\theta} - e^{-\theta}}{e^{\theta} + e^{-\theta}}\)
   
   
5. Relation Between Hyperbolic and Trigonometric Sines/Cosines
   1. \(\cos i\theta= \frac{e^{i^2\theta} + e^{-i^2\theta}}{2}=\frac{e^{\theta} + e^{-\theta}}{2}=\cosh \theta\)
   2. \(\sin i\theta= \frac{e^{i^2\theta} - e^{-i^2\theta}}{2i}=\frac{e^{-\theta} - e^{\theta}}{2i}=\frac{i(e^{\theta} - e^{-\theta})}{2}=i\sinh \theta\hspace{.5cm}\Rightarrow \sinh \theta=-i\sin i\theta\)
   3. \(\tan i\theta= \frac{\sin i\theta}{\cos i\theta}= \frac{i\sinh \theta}{\cosh\theta}=i\tanh \theta\hspace{.5cm}\Rightarrow \tanh \theta=-i\tan i\theta\)
   4. \(\cosh i\theta= \frac{e^{i\theta} + e^{-i\theta}}{2}=\cos \theta\)
   5. \(\sinh i\theta= \frac{i(e^{i\theta} - e^{-i\theta})}{2i}=i\sin \theta\hspace{.5cm}\Rightarrow \sin \theta=-i\sinh i\theta\)
   6. \(\tanh i\theta= \frac{\sinh i\theta}{\cosh i\theta}= \frac{i\sin \theta}{\cos \theta}=i\tan \theta\hspace{.5cm}\Rightarrow \tan \theta=-i\tanh i\theta\)
6. Trigonometric Angle Sum/Difference Identities
   1. \(\sin(A + B) = \sin A \cos B + \cos A \sin B\)
      
   2. \(\sin(A - B) = \sin A \cos B - \cos A \sin B\)
      
   3. \(\cos(A + B) = \cos A \cos B - \sin A \sin B\)
      
   4. \(\cos(A - B) = \cos(B - A)=\cos A \cos B + \sin A \sin B\)
      
   5. \(\tan(A + B) = \frac{\sin(A + B)}{\cos(A + B)}=\frac{\sin A \cos B + \cos A \sin B}{\cos A \cos B - \sin A \sin B}=\frac{\tan A + \tan B}{1 - \tan A \tan B}\)
   6. \(\tan(A - B) = \frac{\sin(A - B)}{\cos(A - B)}=\frac{\sin A \cos B - \cos A \sin B}{\cos A \cos B + \sin A \sin B}=\frac{\tan A - \tan B}{1 + \tan A \tan B}\)
   7. \(\cot(A + B) = \frac{\cos(A + B)}{\sin(A + B)}=\frac{\cos A \cos B - \sin A \sin B}{\sin A \cos B + \cos A \sin B}=\frac{\cot A \cot B - 1}{\cot B + \cot A}\)
   8. \(\cot(A - B) = \frac{\cos(A - B)}{\sin(A - B)}=\frac{\cos A \cos B + \sin A \sin B}{\sin A \cos B - \cos A \sin B}=\frac{\cot A \cot B + 1}{\cot B - \cot A}\)
7. Hyperbolic Angle Sum/Difference Identities
   1. \(\sinh(A + B) = -i\sin(iA + iB)=-i(\sin iA \cos iB + \cos iA \sin iB)=-i(i\sinh A \cosh B + \cosh A\hspace{.2cm}i\sinh B)=\sinh A \cosh B + \cosh A \sinh B\)
   2. \(\sinh(A - B) = -i\sin(iA + iB)=-i(\sin iA \cos iB - \cos iA \sin iB)=-i(i\sinh A \cosh B - \cosh A\hspace{.2cm}i\sinh B) = \sinh A \cosh B - \cosh A \sinh B\)
   3. \(\cosh(A + B) =\cos(iA + iB)=(\cos iA \cos iB - \sin iA \sin iB)=(\cosh A \cosh B - i\sinh A\hspace{.2cm}i\sinh B) = \cosh A \cosh B + \sinh A \sinh B\)
   4. \(\cosh(A - B) =\cosh(B - A)=\cos(iA - iB)=(\cos iA \cos iB + \sin iA \sin iB)=(\cosh A \cosh B + i\sinh A\hspace{.2cm}i\sinh B) = \cosh A \cosh B - \sinh A \sinh B\)
   5. \(\tanh(A + B) = \frac{\sinh(A + B)}{\cosh(A + B)}=\frac{\sinh A \cosh B + \cosh A \sinh B}{\cosh A \cosh B + \sinh A \sinh B}=\frac{\tanh A + \tanh B}{1 + \tanh A \tanh B}\)
   6. \(\tanh(A - B) = \frac{\sinh(A - B)}{\cosh(A - B)}=\frac{\sinh A \cosh B - \cosh A \sinh B}{\cosh A \cosh B - \sinh A \sinh B}=\frac{\tanh A - \tanh B}{1 - \tanh A \tanh B}\)
   7. \(\coth(A + B) = \frac{\cosh(A + B)}{\sinh(A + B)}=\frac{\cosh A \cosh B + \sinh A \sinh B}{\sinh A \cosh B + \cosh A \sinh B}=\frac{\coth A \coth B + 1}{\coth B + \coth A}\)
   8. \(\coth(A - B) = \frac{\cosh(A - B)}{\sinh(A - B)}=\frac{\cosh A \cosh B - \sinh A \sinh B}{\sinh A \cosh B - \cosh A \sinh B}=\frac{\coth A \coth B - 1}{\coth B - \coth A}\)
8. Trigonometric Product to Sum Identities
   1. \(\sin A \cos B=\frac{\sin(A+B)+\sin(A-B)}{2}\)
   2. \(\cos A \sin B=\frac{\sin(A+B)-\sin(A-B)}{2}\)
   3. \(\cos A \cos B=\frac{\cos(A+B)+\cos(A-B)}{2}\)
   4. \(\sin A \sin B=\frac{\cos(A-B)-\cos(A+B)}{2}\)
9. Hyperbolic Product to Sum Identities
   1. \(\sinh A \cosh B=\frac{\sinh(A+B)+\sinh(A-B)}{2}\)
   2. \(\cosh A \sinh B=\frac{\sinh(A+B)-\sinh(A-B)}{2}\)
   3. \(\cosh A \cosh B=\frac{\cosh(A+B)+\cosh(A-B)}{2}\)
   4. \(\sinh A \sinh B=\frac{\cos(A+B)-\cosh(A-B)}{2}\)
10. Trigonometric Sum to Product Identities
    1. \(\sin 2A + \sin 2B=2\sin(A+B)\cos(A-B)\hspace{.5cm}\Rightarrow \sin A + \sin B=2\sin(\frac{A+B}{2})\cos(\frac{A-B}{2})\)
       
       Derivation
       
       \(\sin(A+B)\cos(A-B)=(\sin A \cos B + \cos A \sin B)(\cos A \cos B + \sin A \sin B)\)
       
       \(= {\cos}^2 B \sin A \cos A + {\sin}^2 A \sin B \cos B + {\cos}^2 A \sin B \cos B + {\sin}^2 B \sin A \cos A\)
       
       \(=({\cos}^2 B + {\sin}^2 B) \sin A \cos A + ({\sin}^2 A + {\cos}^2 A)\sin B \cos B\)
       
       \(=\sin A \cos A +\sin B \cos B=\frac{2\sin A \cos A +2\sin B \cos B}{2}=\frac{\sin 2A +\sin 2B}{2} \)
       
       \(\Rightarrow \sin 2A +\sin 2B= 2\sin(A+B)\cos(A-B)\hspace{.5cm}\Rightarrow \sin A +\sin B= 2\sin(\frac{A+B}{2})\cos(\frac{A-B}{2})\)
       
       
    2. \(\sin 2A - \sin 2B=2\sin(A-B)\cos(A+B)\hspace{.5cm}\Rightarrow \sin A - \sin B=2\sin(\frac{A-B}{2})\cos(\frac{A+B}{2})\)
       
       Derivation
       
       \(\sin(A-B)\cos(A+B)=(\sin A \cos B - \cos A \sin B)(\cos A \cos B - \sin A \sin B)\)
       
       \(= {\cos}^2 B \sin A \cos A - {\sin}^2 A \sin B \cos B - {\cos}^2 A \sin B \cos B + {\sin}^2 B \sin A \cos A\)
       
       \(=({\cos}^2 B + {\sin}^2 B) \sin A \cos A - ({\sin}^2 A + {\cos}^2 A)\sin B \cos B\)
       
       \(=\sin A \cos A -\sin B \cos B=\frac{2\sin A \cos A -2\sin B \cos B}{2}=\frac{\sin 2A -\sin 2B}{2} \)
       
       \(\Rightarrow \sin 2A -\sin 2B= 2\sin(A-B)\cos(A+B)\hspace{.5cm}\Rightarrow \sin A -\sin B= 2\sin(\frac{A-B}{2})\cos(\frac{A+B}{2})\)
       
       
    3. \(\cos 2A + \cos 2B=2\cos(A+B)\cos(A-B)\hspace{.5cm}\Rightarrow \cos A + \cos B=2\cos(\frac{A+B}{2})\cos(\frac{A-B}{2})\)
       
       Derivation
       
       \(\cos(A+B)\cos(A-B)=(\cos A \cos B - \sin A \sin B)(\cos A \cos B + \sin A \sin B)\)
       
       \(= {\cos}^2 A {\cos}^2 B + \sin A \sin B \cos A \cos B - \sin A \sin B \cos A \cos B - {\sin}^2 A {\sin}^2 B\)
       
       \(={\cos}^2 A {\cos}^2 B - {\sin}^2 A {\sin}^2 B= (1-{\sin}^2 A) (1-{\sin}^2 B) - {\sin}^2 A {\sin}^2 B\)
       
       \(=1-{\sin}^2 A -{\sin}^2 B + {\sin}^2 A {\sin}^2 B - {\sin}^2 A {\sin}^2 B\)
       
       \(=1-{\sin}^2 A -{\sin}^2 B=\frac{2-2{\sin}^2 A -2{\sin}^2 B}{2}=\frac{(1-2{\sin}^2 A) + (1-2{\sin}^2 B)}{2}=\frac{\cos 2A + \cos 2B}{2}\)
       
       \(\Rightarrow \cos 2A + \cos 2B=2\cos(A+B)\cos(A-B)\hspace{.5cm}\Rightarrow \cos A + \cos B=2\cos(\frac{A+B}{2})\cos(\frac{A-B}{2})\)
       
       
    4. \(\cos 2B - \cos 2A=2\sin(A+B)\sin(A-B)\hspace{.5cm}\Rightarrow \cos B - \cos A=2\sin(\frac{A+B}{2})\sin(\frac{A-B}{2})\)
       
       Derivation
       
       \(\sin(A+B)\sin(A-B)=(\sin A \cos B + \cos A \sin B)(\sin A \cos B - \cos A \sin B)\)
       
       \(= {\sin}^2 A {\cos}^2 B - \sin A \sin B \cos A \cos B + \sin A \sin B \cos A \cos B - {\cos}^2 A {\sin}^2 B\)
       
       \(={\sin}^2 A {\cos}^2 B - {\cos}^2 A {\sin}^2 B= (1-{\cos}^2 A){\cos}^2 B - {\cos}^2 A {\sin}^2 B\)
       
       \(={\cos}^2 B -{\cos}^2 A{\cos}^2 B - {\cos}^2 A {\sin}^2 B={\cos}^2 B-{\cos}^2 A({\sin}^2 B +{\cos}^2 B )\)
       
       \(={\cos}^2 B-{\cos}^2 A=1-{\sin}^2 B -{\cos}^2 A=\frac{2-2{\sin}^2 B -2{\cos}^2 A}{2}=\frac{(1-2{\sin}^2 B) - (2{\cos}^2 A -1)}{2}=\frac{\cos 2B - \cos 2A}{2}\)
       
       \(\Rightarrow \cos 2B - \cos 2A=2\sin(A+B)\sin(A-B)\hspace{.5cm}\Rightarrow \cos B - \cos A=2\sin(\frac{A+B}{2})\sin(\frac{A-B}{2})\)
       
       
11. Trigonometric Identities Similar to Sum to Product Identities
    1. \(\sin(2A + 2B)=2\sin(A+B)\cos(A+B)\)
       
       Derivation
       
       \(\sin(A+B)\cos(A+B)=(\sin A \cos B + \cos A \sin B)(\cos A \cos B - \sin A \sin B)\)
       
       \(= {\cos}^2 B \sin A \cos A - {\sin}^2 A \sin B \cos B + {\cos}^2 A \sin B \cos B - {\sin}^2 B \sin A \cos A\)
       
       \(=({\cos}^2 B - {\sin}^2 B) \sin A \cos A + ({\cos}^2 A - {\sin}^2 A)\sin B \cos B\)
       
       \(=\sin A \cos A \cos 2B+\sin B \cos B \cos 2A=\frac{2\sin A \cos A \cos 2B+2\sin B \cos B \cos 2A}{2}=\frac{\sin 2A \cos 2B +\cos 2A \sin 2B}{2}=\frac{\sin (2A + 2B)}{2}\)
       
       \(\Rightarrow \sin(2A + 2B)=2\sin(A+B)\cos(A+B)\)
       
       
    2. \(\sin(2A - 2B)=2\sin(A-B)\cos(A-B)\)
       
       Derivation
       
       \(\sin(A-B)\cos(A-B)=(\sin A \cos B - \cos A \sin B)(\cos A \cos B + \sin A \sin B)\)
       
       \(= {\cos}^2 B \sin A \cos A + {\sin}^2 A \sin B \cos B - {\cos}^2 A \sin B \cos B - {\sin}^2 B \sin A \cos A\)
       
       \(=({\cos}^2 B - {\sin}^2 B) \sin A \cos A - ({\cos}^2 A - {\sin}^2 A)\sin B \cos B\)
       
       \(=\sin A \cos A \cos 2B-\sin B \cos B \cos 2A=\frac{2\sin A \cos A \cos 2B-2\sin B \cos B \cos 2A}{2}=\frac{\sin 2A \cos 2B -\cos 2A \sin 2B}{2}=\frac{\sin (2A - 2B)}{2}\)
       
       \(\Rightarrow \sin(2A - 2B)=2\sin(A-B)\cos(A-B)\)
       
       
12. Hyperbolic Sum to Product Identities
    1. \(\sinh 2A + \sinh 2B=2\sinh(A+B)\cosh(A-B)\hspace{.5cm}\Rightarrow \sinh A + \sinh B=2\sinh(\frac{A+B}{2})\cosh(\frac{A-B}{2})\)
       
       Derivation
       
       \(\sinh(A+B)\cosh(A-B)=(\sinh A \cosh B + \cosh A \sinh B)(\cosh A \cosh B - \sinh A \sinh B)\)
       
       \(= {\cosh}^2 B \sinh A \cosh A - {\sinh}^2 A \sinh B \cosh B + {\cosh}^2 A \sinh B \cosh B - {\sinh}^2 B \sinh A \cosh A\)
       
       \(=({\cosh}^2 B - {\sinh}^2 B) \sinh A \cosh A + ( {\cosh}^2 A- {\sinh}^2 A)\sinh B \cosh B\)
       
       \(=\sinh A \cosh A +\sinh B \cosh B=\frac{2\sinh A \cosh A +2\sinh B \cosh B}{2}=\frac{\sinh 2A +\sinh 2B}{2} \)
       
       \(\Rightarrow \sinh 2A + \sinh 2B=2\sinh(A+B)\cosh(A-B)\hspace{.5cm}\Rightarrow \sinh A + \sinh B=2\sinh(\frac{A+B}{2})\cosh(\frac{A-B}{2})\)
       
       
    2. \(\sinh 2A - \sinh 2B=2\sinh(A-B)\cosh(A+B)\hspace{.5cm}\Rightarrow \sinh A - \sinh B=2\sinh(\frac{A-B}{2})\cosh(\frac{A+B}{2})\)
       
       Derivation
       
       \(\sinh(A-B)\cosh(A+B)=(\sinh A \cosh B - \cosh A \sinh B)(\cosh A \cosh B + \sinh A \sinh B)\)
       
       \(= {\cosh}^2 B \sinh A \cosh A + {\sinh}^2 A \sinh B \cosh B - {\cosh}^2 A \sinh B \cosh B - {\sinh}^2 B \sinh A \cosh A\)
       
       \(=({\cosh}^2 B - {\sinh}^2 B) \sinh A \cosh A - ( {\cosh}^2 A - {\sinh}^2 A )\sin B \cos B\)
       
       \(=\sinh A \cosh A -\sinh B \cosh B=\frac{2\sinh A \cosh A -2\sinh B \cosh B}{2}=\frac{\sinh 2A -\sinh 2B}{2} \)
       
       \(\Rightarrow \sinh 2A - \sinh 2B=2\sinh(A-B)\cosh(A+B)\hspace{.5cm}\Rightarrow \sinh A - \sinh B=2\sinh(\frac{A-B}{2})\cosh(\frac{A+B}{2})\)
       
       
    3. \(\cosh 2A + \cosh 2B=2\cosh(A+B)\cosh(A-B)\hspace{.5cm}\Rightarrow \cosh A + \cosh B=2\cosh(\frac{A+B}{2})\cosh(\frac{A-B}{2})\)
       
       Derivation
       
       \(\cosh(A+B)\cosh(A-B)=(\cosh A \cosh B + \sinh A \sinh B)(\cosh A \cosh B - \sinh A \sinh B)\)
       
       \(= {\cosh}^2 A {\cosh}^2 B + \sinh A \sinh B \cosh A \cosh B - \sinh A \sinh B \cosh A \cosh B - {\sinh}^2 A {\sinh}^2 B\)
       
       \(={\cosh}^2 A {\cosh}^2 B - {\sinh}^2 A {\sinh}^2 B= (1+{\sinh}^2 A) (1+{\sinh}^2 B) - {\sinh}^2 A {\sinh}^2 B\)
       
       \(=1+{\sinh}^2 A +{\sinh}^2 B + {\sinh}^2 A {\sinh}^2 B - {\sinh}^2 A {\sinh}^2 B\)
       
       \(=1+{\sinh}^2 A +{\sinh}^2 B=\frac{2+2{\sinh}^2 A +2{\sinh}^2 B}{2}=\frac{(1+2{\sinh}^2 A) + (1+2{\sinh}^2 B)}{2}=\frac{\cosh 2A + \cosh 2B}{2}\)
       
       \(\Rightarrow \cosh 2A + \cosh 2B=2\cosh(A+B)\cosh(A-B)\hspace{.5cm}\Rightarrow \cosh A + \cosh B=2\cosh(\frac{A+B}{2})\cosh(\frac{A-B}{2})\)
       
       
    4. \(\cosh 2A - \cosh 2B=2\sinh(A+B)\sinh(A-B)\hspace{.5cm}\Rightarrow \cosh A - \cosh B=2\sinh(\frac{A+B}{2})\sinh(\frac{A-B}{2})\)
       
       Derivation
       
       \(\sinh(A+B)\sinh(A-B)=(\sinh A \cosh B + \cosh A \sinh B)(\sinh A \cosh B - \cosh A \sinh B)\)
       
       \(= {\sinh}^2 A {\cosh}^2 B - \sinh A \sinh B \cosh A \cosh B + \sinh A \sinh B \cosh A \cosh B - {\cosh}^2 A {\sinh}^2 B\)
       
       \(={\sinh}^2 A {\cosh}^2 B - {\cosh}^2 A {\sinh}^2 B= ({\cosh}^2 A-1){\cosh}^2 B - {\cosh}^2 A {\sinh}^2 B\)
       
       \(={\cosh}^2 A{\cosh}^2 B -{\cosh}^2 B - {\cosh}^2 A {\sinh}^2 B={\cosh}^2 A({\cosh}^2 B -{\sinh}^2 B ) -{\cosh}^2 B\)
       
       \(={\cosh}^2 A-{\cosh}^2 B=1+{\sinh}^2 A -{\cosh}^2 B=\frac{2+2{\sinh}^2 A -2{\cosh}^2 B}{2}=\frac{(1+2{\sinh}^2 A) - (2{\cosh}^2 B -1)}{2}=\frac{\cosh 2A - \cosh 2B}{2}\)
       
       \(\Rightarrow \cosh 2A - \cosh 2B=2\sinh(A+B)\sinh(A-B)\hspace{.5cm}\Rightarrow \cosh A - \cosh B=2\sinh(\frac{A+B}{2})\sinh(\frac{A-B}{2})\)
       
       
13. Hyperbolic Identities Similar to Sum to Product Identities
    1. \(\sinh(2A + 2B)=2\sinh(A+B)\cosh(A+B)\)
       
       Derivation
       
       \(\sinh(A+B)\cosh(A+B)=(\sinh A \cosh B + \cosh A \sinh B)(\cosh A \cosh B + \sinh A \sinh B)\)
       
       \(= {\cosh}^2 B \sinh A \cosh A + {\sinh}^2 A \sinh B \cosh B + {\cosh}^2 A \sinh B \cosh B + {\sinh}^2 B \sinh A \cosh A\)
       
       \(=({\cosh}^2 B + {\sinh}^2 B) \sinh A \cosh A + ({\cosh}^2 A + {\sinh}^2 A)\sinh B \cosh B\)
       
       \(=\sinh A \cosh A \cosh 2B+\sin B \cos B \cosh 2A=\frac{2\sinh A \cosh A \cosh 2B+2\sinh B \cosh B \cosh 2A}{2}=\frac{\sinh 2A \cosh 2B +\cosh 2A \sinh 2B}{2}=\frac{\sinh (2A + 2B)}{2}\)
       
       \(\Rightarrow \sinh(2A + 2B)=2\sinh(A+B)\cosh(A+B)\)
       
       
    2. \(\sinh(2A - 2B)=2\sinh(A-B)\cosh(A-B)\)
       
       Derivation
       
       \(\sinh(A-B)\cosh(A-B)=(\sinh A \cosh B - \cosh A \sinh B)(\cosh A \cosh B - \sinh A \sinh B)\)
       
       \(= {\cosh}^2 B \sin A \cos A - {\sinh}^2 A \sinh B \cosh B - {\cosh}^2 A \sinh B \cosh B + {\sinh}^2 B \sinh A \cosh A\)
       
       \(=({\cosh}^2 B + {\sinh}^2 B) \sin A \cos A - ({\cosh}^2 A + {\sinh}^2 A)\sinh B \cosh B\)
       
       \(=\sinh A \cosh A \cosh 2B-\sinh B \cosh B \cosh 2A=\frac{2\sinh A \cosh A \cosh 2B-2\sinh B \cosh B \cosh 2A}{2}=\frac{\sinh 2A \cosh 2B -\cosh 2A \sinh 2B}{2}=\frac{\sinh (2A - 2B)}{2}\)
       
       \(\Rightarrow \sinh(2A - 2B)=2\sinh(A-B)\cosh(A-B)\)
       
       
14. Linear Combinations of Trigonometric Sine and Cosine having Same Frequency
    1. \(A\sin \theta + B\cos \theta=C \sin(\theta + \phi_1)=C \cos(\theta - \phi_2)=C \cos(\phi_2 - \theta)\)
       
       Derivation
       
       \(A\sin \theta + B\cos \theta=\sqrt{A^2 + B^2}(\frac{A}{\sqrt{A^2 + B^2}}\sin\theta + \frac{B}{\sqrt{A^2 + B^2}}\cos\theta)\)
       
       Setting \(C=\sqrt{A^2 + B^2}, \hspace{.2cm}\cos \phi_1=\frac{A}{\sqrt{A^2 + B^2}}, \hspace{.2cm}\sin \phi_1=\frac{B}{\sqrt{A^2 + B^2}}, \hspace{.2cm}\sin \phi_2=\frac{A}{\sqrt{A^2 + B^2}}\) and \(\cos \phi_2=\frac{B}{\sqrt{A^2 + B^2}}\) we have
       
       \(A\sin \theta + B\cos \theta=C (\sin \theta \cos \phi_1 + \cos \theta \sin \phi_1)=C \sin(\theta + \phi_1)\)
       
       \(A\sin \theta + B\cos \theta=C (\sin \theta \sin \phi_2 + \cos \theta \cos \phi_2)=C \cos(\theta - \phi_2)=C \cos(\phi_2 - \theta)\)
       
       
    2. \(A\sin \theta - B\cos \theta=C \sin(\theta - \phi_1)=-C \cos(\theta + \phi_2)\)
       
       Derivation
       
       \(A\sin \theta - B\cos \theta=\sqrt{A^2 + B^2}(\frac{A}{\sqrt{A^2 + B^2}}\sin\theta - \frac{B}{\sqrt{A^2 + B^2}}\cos\theta)\)
       
       Setting \(C=\sqrt{A^2 + B^2}, \hspace{.2cm}\cos \phi_1=\frac{A}{\sqrt{A^2 + B^2}}, \hspace{.2cm}\sin \phi_1=\frac{B}{\sqrt{A^2 + B^2}}, \hspace{.2cm}\sin \phi_2=\frac{A}{\sqrt{A^2 + B^2}}\) and \(\cos \phi_2=\frac{B}{\sqrt{A^2 + B^2}}\) we have
       
       \(A\sin \theta - B\cos \theta=C (\sin \theta \cos \phi_1 - \cos \theta \sin \phi_1)=C \sin(\theta - \phi_1)\)
       
       \(A\sin \theta - B\cos \theta=C (\sin \theta \sin \phi_2 - \cos \theta \cos \phi_2)=-C (\cos \theta \cos \phi_2-\sin \theta \sin \phi_2)=-C \cos(\theta + \phi_2)\)
       
       
    3. \(A\cos \theta - B\sin \theta=C \cos(\theta + \phi_1)=C \sin(\phi_2-\theta)\)=-C \sin(\theta - \phi_2)\)
       
       Derivation
       
       \(A\cos \theta - B\sin \theta=\sqrt{A^2 + B^2}(\frac{A}{\sqrt{A^2 + B^2}}\cos\theta - \frac{B}{\sqrt{A^2 + B^2}}\sin\theta)\)
       
       Setting \(C=\sqrt{A^2 + B^2}, \hspace{.2cm}\cos \phi_1=\frac{A}{\sqrt{A^2 + B^2}}, \hspace{.2cm}\sin \phi_1=\frac{B}{\sqrt{A^2 + B^2}}, \hspace{.2cm}\sin \phi_2=\frac{A}{\sqrt{A^2 + B^2}}\) and \(\cos \phi_2=\frac{B}{\sqrt{A^2 + B^2}}\) we have
       
       \(A\cos \theta - B\sin \theta=C (\cos \theta \cos \phi_1 - \sin \theta \sin \phi_1)=C \cos(\theta + \phi_1)\)
       
       \(A\cos \theta - B\sin \theta=C (\cos \theta \sin \phi_2 - \sin \theta \cos \phi_2)=C \sin(\phi_2-\theta)=-C (\sin \theta \cos \phi_2-\cos \theta \sin \phi_2)=-C \sin(\theta - \phi_2)\)
       
       
    4. \(A_1\sin(\theta + a_1) + A_2\sin(\theta + a_2) + \cdots + A_n\sin(\theta + a_n)=B\sin(\theta + \phi_1)=B\cos(\theta - \phi_2)=B\cos(\phi_2-\theta)\)
       
       Derivation
       
       \(A_1\sin(\theta + a_1) + A_2\sin(\theta + a_2) + \cdots + A_n\sin(\theta + a_n)\)
       
       \(=A_1\sin\theta \cos a_1 + A_1\cos\theta \sin a_1 + A_2\sin\theta \cos a_2 + A_2\cos\theta \sin a_2 + \cdots + A_n\sin\theta \cos a_n + A_n\cos\theta \sin a_n\)
       
       \(=(A_1 \cos a_1 + A_2 \cos a_2 + \cdots + A_n \cos a_n )\sin\theta + (A_1\sin a_1 + A_2\sin a_2 + \cdots + A_n\sin a_n) \cos\theta\)
       
       \(=\sqrt{{A_1}^2 + {A_2}^2 + \cdots + {A_n}^2 + 2A_1A_2 \cos(a_1-a_2) + 2A_1A_3 \cos(a_1-a_3) + \cdots}\)
       \((\frac{A_1 \cos a_1 + A_2 \cos a_2 + \cdots + A_n \cos a_n }{\sqrt{{A_1}^2 + {A_2}^2 + \cdots + {A_n}^2 + 2A_1A_2 \cos(a_1-a_2) + 2A_1A_3 \cos(a_1-a_3) + \cdots}}\sin\theta + \frac{A_1\sin a_1 + A_2\sin a_2 + \cdots + A_n\sin a_n}{\sqrt{{A_1}^2 + {A_2}^2 + \cdots + {A_n}^2 + 2A_1A_2 \cos(a_1-a_2) + 2A_1A_3 \cos(a_1-a_3) + \cdots}} \cos\theta)\)
       
       Setting \(B=\sqrt{{A_1}^2 + {A_2}^2 + \cdots + {A_n}^2 + 2A_1A_2 \cos(a_1-a_2) + 2A_1A_3 \cos(a_1-a_3) + \cdots}\),
       
       \(\cos \phi_1=\frac{A_1 \cos a_1 + A_2 \cos a_2 + \cdots + A_n \cos a_n }{\sqrt{{A_1}^2 + {A_2}^2 + \cdots + {A_n}^2 + 2A_1A_2 \cos(a_1-a_2) + 2A_1A_3 \cos(a_1-a_3) + \cdots}}\),
       
       \(\sin \phi_1=\frac{A_1\sin a_1 + A_2\sin a_2 + \cdots + A_n\sin a_n}{\sqrt{{A_1}^2 + {A_2}^2 + \cdots + {A_n}^2 + 2A_1A_2 \cos(a_1-a_2) + 2A_1A_3 \cos(a_1-a_3) + \cdots}}\) we get
       
       \(\sin \phi_2=\frac{A_1 \cos a_1 + A_2 \cos a_2 + \cdots + A_n \cos a_n }{\sqrt{{A_1}^2 + {A_2}^2 + \cdots + {A_n}^2 + 2A_1A_2 \cos(a_1-a_2) + 2A_1A_3 \cos(a_1-a_3) + \cdots}}\),
       
       \(\cos \phi_2=\frac{A_1\sin a_1 + A_2\sin a_2 + \cdots + A_n\sin a_n}{\sqrt{{A_1}^2 + {A_2}^2 + \cdots + {A_n}^2 + 2A_1A_2 \cos(a_1-a_2) + 2A_1A_3 \cos(a_1-a_3) + \cdots}}\) we get
       
       \(A_1\sin(\theta + a_1) + A_2\sin(\theta + a_2) + \cdots + A_n\sin(\theta + a_n)=B(\sin\theta \cos \phi_1 + \cos\theta\sin\phi_1)=B\sin(\theta + \phi_1)\)
       
       \(A_1\sin(\theta + a_1) + A_2\sin(\theta + a_2) + \cdots + A_n\sin(\theta + a_n)=B(\sin\theta \sin \phi_2 + \cos\theta\cos\phi_2)=B\cos(\theta - \phi_2)=B\cos(\phi_2 - \theta)\)
       
       
    5. \(A_1\cos(\theta + a_1) + A_2\cos(\theta + a_2) + \cdots + A_n\cos(\theta + a_n)=B\cos(\theta + \phi_1)=B\sin(\phi_2-\theta)=-B\sin(\theta-\phi_2)\)
       
       Derivation
       
       \(A_1\cos(\theta + a_1) + A_2\cos(\theta + a_2) + \cdots + A_n\cos(\theta + a_n)\)
       
       \(=A_1\cos\theta \cos a_1 - A_1\sin\theta \sin a_1 + A_2\cos\theta \cos a_2 - A_2\sin\theta \sin a_2 + \cdots + A_n\cos\theta \cos a_n - A_n\sin\theta \sin a_n\)
       
       \(=(A_1 \cos a_1 + A_2 \cos a_2 + \cdots + A_n \cos a_n )\cos\theta - (A_1\sin a_1 + A_2\sin a_2 + \cdots + A_n\sin a_n) \sin\theta\)
       
       \(=\sqrt{{A_1}^2 + {A_2}^2 + \cdots + {A_n}^2 + 2A_1A_2 \cos(a_1-a_2) + 2A_1A_3 \cos(a_1-a_3) + \cdots}\)
       \((\frac{A_1 \cos a_1 + A_2 \cos a_2 + \cdots + A_n \cos a_n }{\sqrt{{A_1}^2 + {A_2}^2 + \cdots + {A_n}^2 + 2A_1A_2 \cos(a_1-a_2) + 2A_1A_3 \cos(a_1-a_3) + \cdots}}\cos\theta - \frac{A_1\sin a_1 + A_2\sin a_2 + \cdots + A_n\sin a_n}{\sqrt{{A_1}^2 + {A_2}^2 + \cdots + {A_n}^2 + 2A_1A_2 \cos(a_1-a_2) + 2A_1A_3 \cos(a_1-a_3) + \cdots}} \sin\theta)\)
       
       Setting \(B=\sqrt{{A_1}^2 + {A_2}^2 + \cdots + {A_n}^2 + 2A_1A_2 \cos(a_1-a_2) + 2A_1A_3 \cos(a_1-a_3) + \cdots}\),
       
       \(\cos \phi_1=\frac{A_1 \cos a_1 + A_2 \cos a_2 + \cdots + A_n \cos a_n }{\sqrt{{A_1}^2 + {A_2}^2 + \cdots + {A_n}^2 + 2A_1A_2 \cos(a_1-a_2) + 2A_1A_3 \cos(a_1-a_3) + \cdots}}\),
       
       \(\sin \phi_1=\frac{A_1\sin a_1 + A_2\sin a_2 + \cdots + A_n\sin a_n}{\sqrt{{A_1}^2 + {A_2}^2 + \cdots + {A_n}^2 + 2A_1A_2 \cos(a_1-a_2) + 2A_1A_3 \cos(a_1-a_3) + \cdots}}\) we get
       
       \(\sin \phi_2=\frac{A_1 \cos a_1 + A_2 \cos a_2 + \cdots + A_n \cos a_n }{\sqrt{{A_1}^2 + {A_2}^2 + \cdots + {A_n}^2 + 2A_1A_2 \cos(a_1-a_2) + 2A_1A_3 \cos(a_1-a_3) + \cdots}}\),
       
       \(\cos \phi_2=\frac{A_1\sin a_1 + A_2\sin a_2 + \cdots + A_n\sin a_n}{\sqrt{{A_1}^2 + {A_2}^2 + \cdots + {A_n}^2 + 2A_1A_2 \cos(a_1-a_2) + 2A_1A_3 \cos(a_1-a_3) + \cdots}}\) we get
       
       \(A_1\cos(\theta + a_1) + A_2\cos(\theta + a_2) + \cdots + A_n\cos(\theta + a_n)=B (\cos \theta \cos \phi_1 - \sin \theta \sin \phi_1)=B\cos(\theta + \phi_1)\)
       
       \(A_1\cos(\theta + a_1) + A_2\cos(\theta + a_2) + \cdots + A_n\cos(\theta + a_n)=B (\cos \theta \sin \phi_2 - \sin \theta \cos \phi_2)=B \sin(\phi_2-\theta)=-B (\sin \theta \cos \phi_2-\cos \theta \sin \phi_2)=-B \sin(\theta - \phi_2)\)