JEE Main 2013 — Trigonometry

A value of $x$ for which $\sin \left(\cot ^{-1}(1+x)\right)=\cos$ $\left(\tan ^{-1} x\right)$, is :

Let $\mathrm{A}=\{\theta: \sin (\theta)=\tan (\theta)\}$ and $\mathrm{B}=(\theta: \cos (\theta)=$ 1\} be two sets. Then:

Let $S=\left\{\left(\begin{array}{ll}a_{11} & a_{12} \\ a_{21} & a_{22}\end{array}\right): a_{i j} \in\{0,1,2\}, a_{11}=a_{22}\right\}$ Then the number of non-singular matrices in the set $S$ is :

The expression $\frac{\mathrm{tanA}}{1-\mathrm{cotA}}+\frac{\mathrm{cotA}}{1-\mathrm{tanA}}$ can be written as :

The number of solutions of the equation $\sin 2 x-2 \cos x+4 \sin x=4$ in the interval $[0,5 \pi]$ is :

The number of solutions of the equation, $\sin ^{-1} x=2 \tan ^{-1} x$ (in principal values) is :

$S=\tan ^{-1}\left(\frac{1}{n^2+n+1}\right)+\tan ^{-1}\left(\frac{1}{n^2+3 n+3}\right)+\ldots$ $+\tan ^{-1}\left(\frac{1}{1+(n+19)(n+20)}\right)$, then $\tan S$ is equal to :