Question

The value of $\tan \left({\sin }^{-1}\left(\cos \left({\sin }^{-1}x\right)\right)\right)\tan \left({\cos }^{-1}\left(\sin \left({\cos }^{-1}x\right)\right)\right)$, where $xϵ(0,1)$, is equal to

• A. 0
• B. 1
• C. $-1$
• D. none of these

Answer 1

Answer: (B)

1

$sin\left(co{s}^{-1}\right(x\left)\right)=sin(\frac{\pi }{2}-si{n}^{-1}(x\left)\right)=cos\left(si{n}^{-1}\right(x\left)\right)=\sqrt{1-x^2}$
Substituting in the above expression we get
$tan\left(si{n}^{-1}\right(cos\left(si{n}^{-1}\right(x\left)\right)\left)\right)tan\left(co{s}^{-1}\right(sin\left(co{s}^{-1}\right(x\left)\right)\left)\right)$
$=tan\left(si{n}^{-1}\right(sin\left(co{s}^{-1}\right(x\left)\right)\left)\right)tan\left(co{s}^{-1}\right(cos\left(si{n}^{-1}\right(x\left)\right)\left)\right)$
$=tan\left(co{s}^{-1}\right(x\left)\right)tan\left(si{n}^{-1}\right(x\left)\right)$
$=tan(\frac{\pi }{2}-si{n}^{-1}(x\left)\right)tan\left(si{n}^{-1}\right(x\left)\right)$
$=cot\left(si{n}^{-1}\right(x\left)\right)tan\left(si{n}^{-1}\right(x\left)\right)$
$=cot\left(\theta \right)tan\left(\theta \right)$
$=1$