IIT-JEE 2010 · Paper 1, Mathematics question 47

First equation. $\tan\theta=\cot5\theta=\tan\left(\frac\pi2-5\theta\right)$, so $6\theta=\frac\pi2+n\pi$:
$$\theta=\pm\frac\pi{12},\ \pm\frac\pi4,\ \pm\frac{5\pi}{12}$$
in $\left(-\frac\pi2,\frac\pi2\right)$; none is a multiple of $\frac\pi5$.

Second equation. $\sin2\theta=1-2\sin^22\theta$ gives $\sin2\theta=\frac12$ or $-1$.

Check. $\sin2\theta$ for the six values: $\pm\frac12,\ \pm1,\ \pm\frac12$. Those that work: $\theta=\frac\pi{12}$ and $\frac{5\pi}{12}$ (value $\frac12$) and $-\frac\pi4$ (value $-1$).

Answer: 3