Question

Evaluate the following:
i. $\sin \left({\cot }^{-1}x\right)$
ii. $\sin (\frac{\pi }{2}-{\sin }^{-1}(-\frac{\sqrt{3}}{2}))$

• A. (i)$\frac{1}{\sqrt{x^2+1}}$, (ii)$\frac{1}{2}$
• B. (i)$\frac{1}{\sqrt{x^2-1}}$, (ii)$\frac{-\sqrt{3}}{2}$
• C. (i)$\frac{-1}{\sqrt{x^2+1}}$, (ii)$\frac{1}{2}$
• D. (i)$\frac{-1}{\sqrt{x^2-1}}$, (ii)$\frac{+\sqrt{3}}{2}$

Answer 1

The correct option is A (i)$\frac{1}{\sqrt{x^2+1}}$, (ii)$\frac{1}{2}$

i) Let ${\cot }^{-1}x=y\Rightarrow x=\cot y\Rightarrow \sin y=\frac{1}{\sqrt{x^2+1}}$
$\therefore \sin \left({\cot }^{-1}x\right)=\frac{1}{\sqrt{x^2+1}}$
ii) $\sin (\frac{\pi }{2}-{\sin }^{-1}\left(\frac{-\sqrt{3}}{2}\right))$
$=\sin \left({\cos }^{-1}\left(\frac{-\sqrt{3}}{2}\right)\right)=\sin \left(\frac{5\pi }{6}\right)=\frac{1}{2}$