ISI2015-DCG-66

Question

If $\sin(\sin^{-1} \frac{2}{5} + \cos ^{-1} x) =1$, then $x$ equals

• A. $1$
• B. $\frac{2}{5}$
• C. $\frac{3}{5}$
• D. None of these

Answer 1

$sin \left (sin^{-1}\ \frac{2}{5} + cos^{-1}\ x \right) =1$

$\implies  \left (sin^{-1}\ \frac{2}{5} + cos^{-1}\ x \right) =sin^{-1}\ 1$

$\implies  sin^{-1}\ \frac{2}{5} + cos^{-1}\ x  =\frac{\pi}{2}$

$\implies  cos^{-1}\ x  =\frac{\pi}{2}- sin^{-1}\ \frac{2}{5}$

$\implies  cos^{-1}\ x  =cos^{-1}\ \frac{2}{5}$          $( \because sin^{-1}\ \frac{2}{5} + cos^{-1}\  \frac{2}{5}=\frac{\pi}{2})$

$\implies x= \frac{2}{5}$

$\therefore$ Option $B.$ is correct