(1)

\begin{align*} \sin\left(x\pm y\right) & =\frac{e^{i\left(x\pm y\right)}-e^{-i\left(x\pm y\right)}}{2i}\\ & =\frac{\left(\cos x+i\sin x\right)\left(\cos y\pm i\sin y\right)-\left(\cos x-i\sin x\right)\left(\cos y\mp i\sin y\right)}{2i}\\ & =\frac{\pm2i\cos x\sin y+2i\sin x\cos y}{2i}\\ & =\sin x\cos y\pm\cos x\sin y \end{align*}

(2)

\begin{align*} \cos\left(x\pm y\right) & =\frac{e^{i\left(x\pm y\right)}+e^{-i\left(x\pm y\right)}}{2}\\ & =\frac{\left(\cos x+i\sin x\right)\left(\cos y\pm i\sin y\right)+\left(\cos x-i\sin x\right)\left(\cos y\mp i\sin y\right)}{2}\\ & =\frac{2\cos x\cos y\mp2\sin x\sin y}{2}\\ & =\cos x\cos y\mp\sin x\sin y \end{align*}

(3)

\begin{align*} \tan\left(x\pm y\right) & =\frac{\sin\left(x\pm y\right)}{\cos\left(x\pm y\right)}\\ & =\frac{\sin x\cos y\pm\cos x\sin y}{\cos x\cos y\mp\sin x\sin y}\\ & =\frac{\tan x\pm\tan y}{1\mp\tan x\tan y} \end{align*} \begin{align*} \tan\left(x\pm y\right) & =\sin\left(x\pm y\right)\cos^{-1}\left(x\pm y\right)\\ & =\sin\left(x\pm y\right)\frac{2\cos\left(x\mp y\right)}{\cos\left(2x\right)+\cos\left(2y\right)}\\ & =\frac{\sin\left(2x\right)+\sin\left(\pm2y\right)}{\cos\left(2x\right)+\cos\left(2y\right)}\\ & =\frac{\sin\left(2x\right)\pm\sin\left(2y\right)}{\cos\left(2x\right)+\cos\left(2y\right)} \end{align*}

(4)

\begin{align*} \sin^{-1}\left(x\pm y\right) & =\frac{1}{\sin\left(x\pm y\right)}\\ & =\frac{\sin\left(x\mp y\right)}{\sin\left(x\pm y\right)\sin\left(x\mp y\right)}\\ & =\frac{2\sin\left(x\mp y\right)}{-\cos\left(2x\right)+\cos\left(\pm2y\right)}\\ & =\frac{2\sin\left(x\mp y\right)}{-\cos\left(2x\right)+\cos\left(2y\right)} \end{align*}

(5)

\begin{align*} \cos^{-1}\left(x\pm y\right) & =\frac{1}{\cos\left(x\pm y\right)}\\ & =\frac{\cos\left(x\mp y\right)}{\cos\left(x\pm y\right)\cos\left(x\mp y\right)}\\ & =\frac{2\cos\left(x\mp y\right)}{\cos\left(2x\right)+\cos\left(2y\right)} \end{align*}

(6)

\begin{align*} \tan^{-1}\left(x\pm y\right) & =\frac{1\mp\tan x\tan y}{\tan x\pm\tan y}\\ & =\frac{\cos x\cos y\mp\sin x\sin y}{\sin x\cos y\pm\cos x\sin y}\\ & =\frac{\frac{\cos x}{\sin x}\cdot\frac{\cos y}{\sin y}\mp1}{\frac{\cos y}{\sin y}\pm\frac{\cos x}{\sin x}}\\ & =\frac{\tan^{-1}x\tan^{-1}y\mp1}{\tan^{-1}y\pm\tan^{-1}x}\\ & =\frac{-1\pm\tan^{-1}x\tan^{-1}y}{\tan^{-1}x\pm\tan^{-1}y} \end{align*} \begin{align*} \tan^{-1}\left(x\pm y\right) & =\cos\left(x\pm y\right)\sin^{-1}\left(x\pm y\right)\\ & =\cos\left(x\pm y\right)\frac{2\sin\left(x\mp y\right)}{-\cos\left(2x\right)+\cos\left(2y\right)}\\ & =\frac{\sin\left(2x\right)+\sin\left(\mp2y\right)}{-\cos\left(2x\right)+\cos\left(2y\right)}\\ & =\frac{\sin\left(2x\right)\mp\sin\left(2y\right)}{-\cos\left(2x\right)+\cos\left(2y\right)} \end{align*}

(1)

\begin{align*} \sinh\left(x\pm y\right) & =-i\sin\left\{ i\left(x\pm y\right)\right\} \\ & =-i\left[\sin\left(ix\right)\cos\left(iy\right)\pm\cos\left(ix\right)\sin\left(iy\right)\right]\\ & =-i\left\{ i\left(\sinh x\right)\left(\cos y\right)\pm\left(\cosh x\right)\left(i\sinh y\right)\right\} \\ & =\sinh x\cosh y\pm\cosh x\sinh y \end{align*}

(2)

\begin{align*} \cosh\left(x\pm y\right) & =\cos\left\{ i\left(x\pm y\right)\right\} \\ & =\cos\left(ix\right)\cos\left(iy\right)\mp\sin\left(ix\right)\sin\left(iy\right)\\ & =\cosh x\cosh y\mp\left(i\sinh x\right)\left(i\sinh y\right)\\ & =\cosh x\cosh y\pm\sinh x\sinh y \end{align*}

(3)

\begin{align*} \tanh\left(x\pm y\right) & =\frac{\sinh\left(x\pm y\right)}{\cosh\left(x\pm y\right)}\\ & =\frac{\sinh x\cosh y\pm\cosh x\sinh y}{\cosh x\cosh y\pm\sinh x\sinh y}\\ & =\frac{\tanh x\pm\tanh y}{1\pm\tanh x\tanh y} \end{align*} \begin{align*} \tanh\left(x\pm y\right) & =-i\tan\left(i\left(x\pm y\right)\right)\\ & =-i\frac{\sin\left(2xi\right)\pm\sin\left(2yi\right)}{\cos\left(2xi\right)+\cos\left(2yi\right)}\\ & =\frac{\sinh\left(2x\right)\pm\sinh\left(2y\right)}{\cosh\left(2x\right)+\cosh\left(2y\right)} \end{align*}

(4)

\begin{align*} \sinh^{-1}\left(x\pm y\right) & =i\sin^{-1}\left(i\left(x\pm y\right)\right)\\ & =i\frac{2\sin\left(ix\mp iy\right)}{-\cos\left(2xi\right)+\cos\left(2yi\right)}\\ & =\frac{-2\sinh\left(x\mp y\right)}{-\cosh\left(2x\right)+\cosh\left(2y\right)}\\ & =\frac{2\sinh\left(x\mp y\right)}{\cosh\left(2x\right)-\cosh\left(2y\right)} \end{align*}

(5)

\begin{align*} \cosh^{-1}\left(x\pm y\right) & =\cos^{-1}\left(i\left(x\pm y\right)\right)\\ & =\frac{2\cos\left(i\left(x\mp y\right)\right)}{\cos\left(2xi\right)+\cos\left(2yi\right)}\\ & =\frac{2\cosh\left(x\mp y\right)}{\cosh\left(2x\right)+\cosh\left(2y\right)} \end{align*}

(6)

\begin{align*} \tanh^{-1}\left(x\pm y\right) & =i\tan^{-1}\left(i\left(x\pm y\right)\right)\\ & =i\tan^{-1}\left(ix\pm iy\right)\\ & =i\frac{-1\pm\tan^{-1}\left(ix\right)\tan^{-1}\left(iy\right)}{\tan^{-1}\left(ix\right)\pm\tan^{-1}\left(iy\right)}\\ & =i\frac{-1\pm\left(-i\right)\tanh^{-1}x\cdot\left(-i\right)\tanh^{-1}y}{\left(-i\right)\tanh^{-1}x\pm\left(-i\right)\tanh^{-1}y}\\ & =-\frac{-1\mp\tanh^{-1}x\cdot\tanh^{-1}y}{\tanh^{-1}x\pm\tanh^{-1}y}\\ & =\frac{1\pm\tanh^{-1}x\cdot\tanh^{-1}y}{\tanh^{-1}x\pm\tanh^{-1}y} \end{align*} \begin{align*} \tanh^{-1}\left(x\pm y\right) & =i\tan^{-1}\left(i\left(x\pm y\right)\right)\\ & =i\frac{\sin\left(2xi\right)\mp\sin\left(2yi\right)}{-\cos\left(2xi\right)+\cos\left(2yi\right)}\\ & =\frac{\sinh\left(2x\right)\mp\sinh\left(2y\right)}{\cosh\left(2x\right)-\cosh\left(2y\right)} \end{align*}