(1)

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

(3)

\begin{align*} \tanh(x\pm y) & =\frac{\sinh(x\pm y)}{\cosh(x\pm y)}\\ & =\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\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*}