(1)

\begin{align*} \sin2x & =\sin(x+x)\\ & =\sin x\cos x+\cos x\sin x\\ & =2\sin x\cos x \end{align*}

(2)

\begin{align*} \cos2x & =\cos(x+x)\\ & =\cos x\cos x-\sin x\sin x\\ & =\cos^{2}x-\sin^{2}x\tag{*}\\ & =2\cos^{2}x-1\tag{*}\\ & =1-2\sin^{2}x\tag{*} \end{align*}

(3)

\begin{align*} \tan2x & =\tan(x+x)\\ & =\frac{\tan x+\tan x}{1-\tan x\tan x}\\ & =\frac{2\tan x}{1-\tan^{2}x} \end{align*}

(1)

\begin{align*} \sinh2x & =-i\sin(2ix)\\ & =-2i\sin(ix)\cos(ix)\\ & =2\sinh x\cosh x \end{align*}

(2)

\begin{align*} \cosh2x & =\cos(2ix)\\ & =2\cos^{2}(ix)-1\\ & =2\cosh^{2}x-1\tag{*}\\ & =1+2\sinh^{2}x\tag{*}\\ & =\cosh^{2}x+\sinh^{2}x\tag{*} \end{align*}

(3)

\begin{align*} \tanh2x & =-i\tan(2ix)\\ & =-i\frac{2\tan(ix)}{1-\tan^{2}(ix)}\\ & =\frac{2\tanh x}{1+\tanh^{2}x} \end{align*}

(1)

\begin{align*} \sin3x & =\sin(x+2x)\\ & =\sin x\cos2x+\cos x\sin2x\\ & =\sin x(1-2\sin^{2}x)+\cos x(2\sin x\cos x)\\ & =3\sin x-4\sin^{3}x \end{align*}

(2)

\begin{align*} \cos3x & =\cos(x+2x)\\ & =\cos x\cos2x-\sin x\sin2x\\ & =\cos x(2\cos^{2}x-1)-\sin x(2\sin x\cos x)\\ & =4\cos^{3}x-3\cos x \end{align*}

(3)

\begin{align*} \tan3x & =\frac{\sin3x}{\cos3x}\\ & =\frac{3\sin x-4\sin^{3}x}{4\cos^{3}x-3\cos x}\\ & =\frac{3\tan x\cos^{-2}x-4\tan^{3}x}{4-3\cos^{-2}x}\\ & =\frac{3\tan x(1+\tan^{2}x)-4\tan^{3}x}{4-3(1+\tan^{2}x)}\\ & =\frac{3\tan x-\tan^{3}x}{1-3\tan^{2}x} \end{align*}

(1)

\begin{align*} \sinh3x & =-i\sin3ix\\ & =-i\left\{ 3\sin(ix)-4\sin^{3}(ix)\right\} \\ & =3\sinh x+4\sinh^{3}x \end{align*}

(2)

\begin{align*} \cosh3x & =\cos(3ix)\\ & =4\cos^{3}(ix)-3\cos(ix)\\ & =4\cosh^{3}x--3\cosh x \end{align*}

(3)

\begin{align*} \tanh3x & =-i\tan(3ix)\\ & =-i\frac{3\tan(ix)-\tan^{3}(ix)}{1-3\tan^{2}(ix)}\\ & =\frac{3\tanh x+\tanh^{3}x}{1+3\tanh^{2}x} \end{align*}