Question

$\alpha ={\sin }^{-1}\left(\cos \left({\sin }^{-1}x\right)\right)$ and $\beta ={\cos }^{-1}\left(\sin \left({\cos }^{-1}x\right)\right)$ then:

• A. $\tan \alpha =\cot \beta$
• B. $\tan \alpha =-\cot \beta$
• C. $\tan \alpha =\tan \beta$
• D. $\tan \alpha =-\tan \beta$

Answer 1

The correct option is A $\tan \alpha =\cot \beta$

$\alpha =si{n}^{-1}\left(cos\right(si{n}^{-1}\left(x\right)\left)\right))$
$=\frac{\pi }{2}-co{s}^{-1}\left(cos\right(\frac{\pi }{2}-co{s}^{-1}\left(x\right)\left)\right)$
$=\frac{\pi }{2}-co{s}^{-1}\left(sin\right(co{s}^{-1}\left(x\right)\left)\right)$
$=\frac{\pi }{2}-\beta$
Hence $\alpha =\frac{\pi }{2}-\beta$
Hence $tan\left(\alpha \right)=cot\left(\beta \right)$