Question

The value of $si{n}^{-1}\left\{cot(si{n}^{-1}\sqrt{\left(\frac{2-\sqrt{3}}{4}\right)}+co{s}^{-1}\frac{\sqrt{12}}{4}+se{c}^{-1}\sqrt{2})\right\}$ is:

• A. $0$
• B. $\frac{\pi }{4}$
• C. $\frac{\pi }{6}$
• D. $\frac{\pi }{2}$

Answer 1

The correct option is A $0$

$co{s}^{-1}\frac{\sqrt{12}}{4}=co{s}^{-1}\frac{\sqrt{3}}{2}=\frac{\pi }{6},se{c}^{-1}\sqrt{2}=\frac{\pi }{4}$
$si{n}^{-1}\sqrt{\frac{2-\sqrt{3}}{4}}=si{n}^{-1}\sqrt{\frac{4-2\sqrt{3}}{8}}=si{n}^{-1}\frac{\sqrt{3}-1}{2\sqrt{2}}=\frac{\pi }{12}$
$\therefore$ $si{n}^{-1}\sqrt{\frac{2-\sqrt{3}}{4}}+co{s}^{-1}\frac{\sqrt{12}}{4}+se{c}^{-1}\sqrt{2}=\frac{\pi }{2}$
and $si{n}^{-1}\left(cot\frac{\pi }{2}\right)=si{n}^{-1}0=0$