The equation $\sin x \cos x = 2$ has

The value of $\frac{\sin^2 \theta}{\sin \theta - \cos \theta} - \frac{\sin \theta + \cos \theta}{\tan^2 \theta - 1}$ for all permissible values of $\theta$ is:

$\sum_{k=1}^3 \cos^2 \left((2k-1) \frac{\pi}{12}\right)$ is equal to

The value of $\tan 20^{\circ} \tan 80^{\circ} \cot 50^{\circ} = $

Let $P(\alpha, \beta)$ and $Q(\gamma, \delta)$ be two points that lie on the curve $\tan^2(x+y) + \cos^2(x+y) + y^2 + 2y = 0$ in the $XY$-plane. If the distance between $P$ and $Q$ is $d$,then $\cos d =$

The value of $\log _{10} \tan 1^{\circ} + \log _{10} \tan 2^{\circ} + \dots + \log _{10} \tan 89^{\circ}$ is equal to :-