Q.16.
\mathbf{If} \quad x=\operatorname{cosec}\left(\tan^{-1}\left(\cos\left(\cot^{-1}\left(\sec\left(\sin^{-1}a\right)\right)\right)\right)\right), \quad a \in [0,1] \text{A.~} x^{2}-a^{2}=3 \text{B.~} x^{2}+a^{2}=3 \text{C.~} x^{2}-a^{2}=2 \text{D.~} x^{2}+a^{2}=2 \text{Answer: B} \text{Explanation:} x=\operatorname{cosec}\left(\tan^{-1}\left(\cos\left(\cot^{-1}\left(\sec\left(\sin^{-1}a\right)\right)\right)\right)\right) =\operatorname{cosec}\left(\tan^{-1}\left(\cos\left(\cot^{-1}\left(\sec\left(\sec^{-1}\frac{1}{\sqrt{1-a^{2}}}\right)\right)\right)\right)\right) =\operatorname{cosec}\left(\tan^{-1}\left(\cos\left(\cot^{-1}\left(\frac{1}{\sqrt{1-a^{2}}}\right)\right)\right)\right) =\operatorname{cosec}\left(\tan^{-1}\left(\cos\left(\cos^{-1}\frac{1}{\sqrt{2-a^{2}}}\right)\right)\right) =\operatorname{cosec}\left(\tan^{-1}\left(\frac{1}{\sqrt{2-a^{2}}}\right)\right) =\operatorname{cosec}\left(\operatorname{cosec}^{-1}\left(\sqrt{3-a^{2}}\right)\right) \therefore \quad x=\sqrt{3-a^{2}} \therefore \quad x^2 + a^2 = 3