The value of $ tan \begin{Bmatrix} sin^{-1}(cos(sin^{-1}x))\end{Bmatrix} tan \begin{Bmatrix} cos^{-1} (sin(cos^{-1}x))\end{Bmatrix}$, where $ 0 < x < \pi / 2$, is equal to
Answer & explanation
Correct answer: option 2
We have,
$cos(sin^{-1}x) = \sqrt{1-x^2} $ and $ sin (cos^{-1}=\sqrt{1-x^2}$
$∴ tan \begin{Bmatrix}sin^{-1} \left(cos(sin^{-1}x)\right)\end{Bmatrix} tan \begin{Bmatrix}cos^{-1}\left(sin(cos^{-1}x)\right)\end{Bmatrix}$
$ = tan \begin{Bmatrix}sin^{-1}(\sqrt{1-x^2})\end{Bmatrix} tan \begin{Bmatrix}cos^{-1}(\sqrt{1-x^2}) \end{Bmatrix}$
$ = tan \begin{pmatrix} sin^{-1}\sqrt{1-x^2}\end{pmatrix} tan \begin{Bmatrix}\frac{\pi}{2}-sin^{-1}(\sqrt{1-x^2})\end{Bmatrix}$
$ = tan \begin{pmatrix} sin^{-1}\sqrt{1-x^2}\end{pmatrix} cot \begin{pmatrix} sin^{-1}\sqrt{1-x^2}\end{pmatrix}=1$