Let,$S=\left\{\theta \in[0,2 \pi]: 8^{2 \sin ^{2} \theta}+8^{2 \cos ^{2} \theta}=16\right\}$. Then $n ( S )+\sum_{\theta \in S}\left(\sec \left(\frac{\pi}{4}+2 \theta\right) \operatorname{cosec}\left(\frac{\pi}{4}+2 \theta\right)\right)$ is equal to.

The set of values of $x$ for which the expression $\frac{{\tan 3x - \tan 2x}}{{1 + \tan 3x\tan 2x}} = 1$, is

If $5{\cos ^2}\theta + 7{\sin ^2}\theta - 6 = 0$, then the general value of $\theta $ is

In a triangle $P Q R, P$ is the largest angle and $\cos P=\frac{1}{3}$. Further the incircle of the triangle touches the sides $P Q, Q R$ and $R P$ at $N, L$ and $M$ respectively, such that the lengths of $P N, Q L$ and $R M$ are consecutive even integers. Then possible length$(s)$ of the side$(s)$ of the triangle is (are)

$(A)$ $16$ $(B)$ $18$ $(C)$ $24$ $(D)$ $22$

If the equation $\cos ^{4} \theta+\sin ^{4} \theta+\lambda=0$ has real solutions for $\theta,$ then $\lambda$ lies in the interval

If$\cos 6\theta + \cos 4\theta + \cos 2\theta + 1 = 0$, where $0 < \theta < {180^o}$, then $\theta  =$