Let $f(x) = \tan^{-1} (\cot x - 2 \cot 2x)$. Then $\left[ \sum_{r = 1}^7 f(r) \right]$ is equal to (where $[.]$ represents the greatest integer function).

$\left[\sin \left(\tan ^{-1} \frac{3}{4}\right)\right]^{2}+\left[\sin \left(\tan ^{-1} \frac{4}{3}\right)\right]^{2}=$

Which pair$(s)$ of function$(s)$ is/are equal? (where ${x}$ and $[x]$ denote the fractional part and integral part functions respectively.)

The number of positive integral solutions of $\tan ^{-1} x+\cos ^{-1}\left(\frac{y}{\sqrt{1+y^2}}\right)=\sin ^{-1}\left(\frac{3}{\sqrt{10}}\right)$ are

Assume that $3.13 \leq \pi \leq 3.15$. The integer closest to the value of $\sin ^{-1}(\sin 1 \cos 4+\cos 1 \sin 4)$,where $1$ and $4$ appearing in $\sin$ and $\cos$ are given in radians,is

The number of solutions of the equation $\sin \left[2 \cos^{-1} \left\{\cot \left(2 \tan^{-1} x\right)\right\}\right] = 0$ is