(1)

\begin{align*} \sin(-x) & =\frac{e^{-ix}-e^{ix}}{2i}\\ & =-\frac{e^{ix}-e^{-ix}}{2i}\\ & =-\sin x \end{align*}

(2)

\begin{align*} \cos(-x) & =\frac{e^{-ix}+e^{ix}}{2}\\ & =\cos x \end{align*}

(3)

\begin{align*} \tan(-x) & =\frac{\sin(-x)}{\cos(-x)}\\ & =\frac{-\sin(x)}{\cos x}\\ & =-\tan x \end{align*}

(4)

\begin{align*} \sin\left(\frac{\pi}{2}-x\right) & =\sin\frac{\pi}{2}\cos x-\cos\frac{\pi}{2}\sin x\\ & =\cos x \end{align*}

(5)

\begin{align*} \cos\left(\frac{\pi}{2}-x\right) & =\cos\frac{\pi}{2}\cos x+\sin\frac{\pi}{2}\sin x\\ & =\sin x \end{align*}

(6)

\begin{align*} \tan\left(\frac{\pi}{2}-x\right) & =\frac{\sin\left(\frac{\pi}{2}-x\right)}{\cos\left(\frac{\pi}{2}-x\right)}\\ & =\frac{\cos x}{\sin x}\\ & =\cot x \end{align*}

(7)

\begin{align*} \sin\left(\pi-x\right) & =\sin\pi\cos x-\cos\pi\sin x\\ & =\sin x \end{align*}

(8)

\begin{align*} \cos\left(\pi-x\right) & =\cos\pi\cos x+\sin\pi\sin x\\ & =-\cos x \end{align*}

(9)

\begin{align*} \tan\left(\pi-x\right) & =\frac{\sin(\pi-x)}{\cos(\pi-x)}\\ & =\frac{\sin x}{-\cos x}\\ & =-\tan x \end{align*}