$sin 3\theta = 4 sin\, \theta \,sin \,2\theta \,sin \,4\theta$ in $0\, \le \,\theta\, \le \, \pi$ has :

The smallest positive values of $x$ and $y$ which satisfy $\tan (x - y) = 1,\,$ $\sec (x + y) = \frac{2}{{\sqrt 3 }}$ are

Let $S=\left\{\theta \in[-\pi, \pi]-\left\{\pm \frac{\pi}{2}\right\}: \sin \theta \tan \theta+\tan \theta=\sin 2 \theta\right\} \text {. }$  If $T =\sum_{\theta \in S } \cos 2 \theta$, then $T + n ( S )$ is equal

If $0 \le x \le \pi $ and ${81^{{{\sin }^2}x}} + {81^{{{\cos }^2}x}} = 30$, then $x =$

Let $f(x)=\cos 5 x+A \cos 4 x+B \cos 3 x$ $+C \cos 2 x+D \cos x+E$, and

$T=f(0)-f\left(\frac{\pi}{5}\right)+f\left(\frac{2 \pi}{5}\right)-f\left(\frac{3 \pi}{5}\right)+\ldots+f\left(\frac{8 \pi}{5}\right)-f\left(\frac{9 \pi}{5}\right) \text {. }$Then, $T$

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