Question

If $cosec\left({\tan }^{-1}\left(\cos \left({\cot }^{-1}\left(\sec \left({\sin }^{-1}a\right)\right)\right)\right)\right)=\sqrt{p-a^q}$, where $aϵ[0,1]$.
Find the value of $p+q$

• A. 3
• B. 4
• C. 5
• D. 6

Answer 1

The correct option is C 5

Let $x=cosec\left({\tan }^{-1}\left(\cos \left(\sec \left({\sin }^{-1}a\right)\right)\right)\right)$
$=cosec\left({\tan }^{-1}\left(\cos \left({\cot }^{-1}\left(\frac{1}{\sqrt{1-a^2}}\right)\right)\right)\right)$
$=cosec\left({\tan }^{-1}\left(\frac{1}{\sqrt{2-a^2}}\right)\right)$ $=\sqrt{3-a^2}$

$\therefore p+q=3+2=5$