Question

$\sin (\cot^{-1}(\tan\,\cos^{-1}x)$ is equal to

• A. $x$
• B. $\sqrt{1-x^2}$
• C. $\frac{1}{x}$
• D. none of these

Answer 1

Answer: (A)

$x$

Answer 2

Put $cos^{-1} x= \theta$ $\therefore x= cos \,\theta$ $\therefore tan\, \theta = \frac{\sqrt{1-x^{2}}}{x}$ $\therefore tan\left(cos^{-1} x\right) = tan\, \theta = \frac{\sqrt{1-x^{2}}}{x}$ $\therefore cot^{-1} \left(tan\left(cos^{-1}\right)\right) = cot^{-1}\left(\frac{\sqrt{1-x^{2}}}{x}\right) = \phi$ (say) $\therefore cot\, \phi = \frac{\sqrt{1-x^{2}}}{x}$ $\therefore sin\, \phi = \frac{x}{1}$ $\therefore sin(cot^{-1}\left(tan\left(cos^{-1}\,x\right)\right)$ $= sin \,\phi = x$