Question

The principal value of ${\cos }^{-1}\left[\frac{1}{\sqrt{2}}(\cos \left(\frac{9\pi }{10}\right)-\sin \left(\frac{9\pi }{10}\right))\right]$ is

• A. $\frac{3\pi }{20}$
• B. $\frac{7\pi }{20}$
• C. $\frac{7\pi }{10}$
• D. none of these

Answer 1

The correct option is A $\frac{3\pi }{20}$

Given: ${cos}^{-1}\left(\frac{1}{\sqrt{2}}\right(cos\frac{9\pi }{10}-sin\frac{9\pi }{10}\left)\right)$

Since$\frac{1}{\sqrt{2}}=cos\frac{\pi }{4}=sin\frac{\pi }{4}$

Putting these values above to $b=makeaproperty$

$⟹{cos}^{-1}(cos\frac{\pi }{4}cos\frac{9\pi }{10}-sin\frac{\pi }{4}sin\frac{9\pi }{10})$

As we know that $cosAcosB-sinAsinB=cos(A+B$

$⟹{cos}^{-1}\left(cos\right(\frac{\pi }{4}+\frac{9\pi }{10}\left)\right)$

$⟹{cos}^{-1}\left(cos\frac{23\pi }{20}\right)$

$⟹{cos}^{-1}\left(cos\frac{3\pi }{20}\right)$

$⟹\frac{3\pi }{20}$

Hence the correct option is (A)