Question

Verify this identity: show work cot(x-pi/2)= -tanx

Answer 1

$-tan(x) = cot(x - \frac{\pi}{2})$
$-tan(x) = \frac{cos(x - \frac{\pi}{2})}{sin(x - \frac{\pi}{2})}$
$-tan(x) = \frac{sin(x)}{-cos(x)}$
$-tan(x) = -tan(x)$

There :).

The most important thing to note here is that $cos(x-\frac{\pi}{2}) = sin(x)$
and $sin(x-\frac{\pi}{2}) = -cos(x)$

Normally, over time you end up memorizing these two identities since you use them so often, but proving them is very easy:

$cos(x-\frac{\pi}{2}) = cos(x)cos(\frac{\pi}{2}) + sin(x)sin(\frac{\pi}{2}) = cos(x)*0 + sin(x) * 1$
$= sin(x)$

$sin(x-\frac{\pi}{2}) = sin(x)cos(\frac{\pi}{2}) - cos(x)sin(\frac{\pi}{2}) = sin(x) * 0 - cos(x) * 1$
$= -cos(x)$