Answer
$\tan(x-\frac{\pi}{2})=-\cot x$
Work Step by Step
Start with the left side:
$\tan(x-\frac{\pi}{2})$
Write tangent as sine divided by cosine:
$=\frac{\sin(x-\frac{\pi}{2})}{\cos(x-\frac{\pi}{2})}$
Expand using subtraction formulas for sine and cosine:
$=\frac{\sin x\cos\frac{\pi}{2}-\cos x\sin\frac{\pi}{2}}{\cos x\cos\frac{\pi}{2}+\sin x\sin\frac{\pi}{2}}$
Evaluate $\cos \frac{\pi}{2}$ and $\sin \frac{\pi}{2}$:
$=\frac{\sin x*0-\cos x*1}{\cos x*0+\sin x*1}$
Simplify:
$=\frac{-\cos x}{\sin x}$
$=-\cot x$
Since this equals the right side, the identity has been proven.
