Question

If $\alpha ={\sin }^{-1}\left(\cos \left({\sin }^{-1}x\right)\right)$ and $\beta ={\cos }^{-1}\left(\sin \left({\cos }^{-1}x\right)\right)$, then find $\tan \alpha \tan \beta$

• A. 0
• B. -1
• C. 1
• D. None of these

Answer 1

Answer: (C)

1

$\beta ={\cos }^{-1}\left(\sin (\frac{\pi }{2}-{\sin }^{-1}x)\right)={\cos }^{-1}\left[\cos \left({\sin }^{-1}x\right)\right]$
also $\alpha ={\sin }^{-1}\left(\cos \left({\sin }^{-1}x\right)\right)$
$\Rightarrow \alpha +\beta =\frac{\pi }{2},$ Using Identity $[{\sin }^{-1}x+{\cos }^{-1}x=\frac{\pi }{2}]$
$\Rightarrow$ $\tan \alpha =\cot \beta$
$\tan \alpha \tan \beta =1$