Question

Verify the identity. cot(theta-pi/2)=-tan(theta)

Answer 1

$\cot\left(\theta-\dfrac{\pi}{2}\right)=-\tan(\theta)\\\\L=\dfrac{\cos\left(\theta-\frac{\pi}{2}\right)}{\sin\left(\theta-\frac{\pi}{2}\right)}=\dfrac{\cos\theta\cos\frac{\pi}{2}+\sin\theta\sin\frac{\pi}{2}}{\sin\theta\cos\frac{\pi}{2}-\sin\frac{\pi}{2}\cos\theta}=\dfrac{\cos\theta\cdot0+\sin\theta\cdot1}{\sin\theta\cdot0-1\cdot\cos\theta}\\\\=\dfrac{\sin\theta}{-\cos\theta}=-\tan\theta=R\\\\Used:$

$\\\\\cos(a-b)=\cos(a)\cos(b)+\sin(a)\sin(b)\\\sin(a-b)=\sin(a)\cos(b)-\sin(b)\cos(a)\\\tan(x)=\dfrac{\sin(x)}{\cos(x)}\\\sin\frac{\pi}{2}=1\\\cos\frac{\pi}{2}=0$