Solving trigonometric equation $-2\csc^2 x\cot x=2\csc x$

Question

I'm having a serious brain lag and can't figure out how to solve this equation:

$$-2\csc^2 x\cot x=2\csc x$$

My initial thoughts was that it could be a quadratic in disguise, but that doesn't seem to be it. How do I solve it?

Answer 1

It is a quadratic in disguise.

Since $\csc x$ is never $0$, we can "cancel" and arrive at the equation $\csc x\cot x=-1$. In terms of the more familiar sines and cosines, we have the equation $$\frac{1}{\sin x} \cdot\frac{\cos x}{\sin x}=-1.$$ This is equivalent to $\cos x=-\sin^2 x$. Replace $-\sin^2 x$ by $\cos^2 x-1$. We end up with the quadratic equation $\cos^2 x-\cos x-1=0$ in $\cos x$. The Golden Number strikes again.