The value of $\tan 9^{\circ} - \tan 27^{\circ} - \tan 63^{\circ} + \tan 81^{\circ}$ is $............$.

The number of points of intersection of the two curves $y = 2\sin x$ and $y = 5x^2 + 2x + 3$ is

If $\tan A - \tan B = x$ and $\cot B - \cot A = y,$ then $\cot (A - B) = $

If $5(\tan^2 x - \cos^2 x) = 2\cos 2x + 9$,then $\cos 4x$ 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 $\cos \theta = \frac{\cos \alpha - \cos \beta}{1 - \cos \alpha \cos \beta}$,then one of the values of $\tan \frac{\theta}{2}$ is