I would like to solve the following using only trig identities.
$ \lim_{x \to \pi} {\cot2(x-\pi)}{\cot(x-\frac\pi2)} $
I have so far that the above is equal to $ \lim_{x \to \pi} \frac{\cos2(x-\pi)}{\sin2(x-\pi)}\frac{\cos(x-\frac\pi2)}{\sin(x-\frac\pi2)} = \lim_{x \to \pi} \frac{-\cos2x}{-\sin2x}\frac{-\sin x}{-\cos x} = \lim_{x \to \pi} \frac{\cos2x}{2\sin x\cos x}\frac{\sin x}{\cos x} = \frac12\lim_{x \to \pi} \frac{\cos2x}{\cos x}\frac{1}{\cos x}=\frac12 $
However, I am afraid the correct answer is $-\frac12$. Where am I going wrong? Also, if there is another method of solving this, I would be thankful for the insight.