The number of solutions of the equation $2 \cos(e^x) = 5^x + 5^{-x}$ is:

If $15 \sin^{4} \alpha + 10 \cos^{4} \alpha = 6$ for some $\alpha \in R$,then the value of $27 \sec^{6} \alpha + 8 \operatorname{cosec}^{6} \alpha$ is equal to ....... .

Let $S = \{\theta \in [0, 2\pi] : 8^{2 \sin^2 \theta} + 8^{2 \cos^2 \theta} = 16\}$. 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.

If $\sin \theta + \sin^2 \theta = 1$ and $\cos^{12} \theta + a \cos^{10} \theta + b \cos^8 \theta + c \cos^6 \theta + d = 0$,then:

If $\sin (y+z-x), \sin (z+x-y)$ and $\sin (x+y-z)$ are in $A$.$P$.,then

If $\sec (\theta+\alpha), \sec \theta$ and $\sec (\theta-\alpha)$ are in arithmetic progression,then $\sin ^2 \theta=$