The number of distinct solutions of the equation $\log _{\frac{1}{2}}|\sin x|=2-\log _{\frac{1}{2}}|\cos x|$ in the interval $[0,2 \pi],$ is

The number of solutions of $\sin ^{7} x+\cos ^{7}=1, x \in[0,4 \pi]$ is equal to :

Let $S=\{\theta \in[0,2 \pi): \tan (\pi \cos \theta)+\tan (\pi \sin \theta)=0\}$.

Then $\sum_{\theta \in S } \sin ^2\left(\theta+\frac{\pi}{4}\right)$ is equal to

If $m$ and $n$ respectively are the numbers of positive and negative value of $\theta$ in the interval $[-\pi, \pi]$ that satisfy the equation $\cos 2 \theta \cos \frac{\theta}{2}=\cos 3 \theta \cos \frac{9 \theta}{2}$, then $mn$ is equal to $.............$.

Find the general solution of the equation $\sec ^{2} 2 x=1-\tan 2 x$

If $\cot (\alpha + \beta ) = 0,$ then $\sin (\alpha + 2\beta ) = $