Solve cos(2x)/sin(x) = c

Question

This little problem is the simplified version of a physics problem that I am solving for my physics 2300 class at osu. I cannot seem to be able to isolate x to find a solution to this problem

Thank for helping in advance

Answer 1

Hint: Write $$\frac{\cos(2x)}{\sin(x)}=\frac{1-2\sin^2(x)}{\sin(x)}=c$$ and solve a quadratic. Substituting $$\sin(x)=t$$ we get the solutions $$\left\{\left\{t\to \frac{1}{4} \left(-\sqrt{c^2+8}-c\right)\right\},\left\{t\to \frac{1}{4} \left(\sqrt{c^2+8}-c\right)\right\}\right\}$$

Answer 2

Hint: $\cos(2x)=\cos^2x-\sin^2x=1-2\sin^2x$.