Question

The value of $\sec [{\sin }^{-1}[-\sin \frac{50\pi }{9}]+{\cos }^{-1}\cos [-\frac{31\pi }{9}]]$ is equal to -

• A. $\sec \frac{14\pi }{9}$
• B. $\sec \frac{2\pi }{9}$
• C. $\sqrt{2}$
• D. $-$1

Answer 1

Answer: (D)

$-$1

$\sec [{\sin }^{-1}[-\sin \frac{50\pi }{9}]+{\cos }^{-1}\cos [-\frac{31\pi }{9}]]$

$=\sec [-{\sin }^{-1}\left[\sin \frac{50\pi }{9}\right]+{\cos }^{-1}\cos \left[\frac{31\pi }{9}\right]]$ ..... $[\because {\sin }^{-1}(-x)=-{\sin }^{-1}x\text{and}\cos (-x)=-\cos x]$

$=\sec [-{\sin }^{-1}\left[\sin (\frac{54\pi }{9}-\frac{4\pi }{9})\right]+{\cos }^{-1}\cos [\frac{27\pi }{9}+\frac{4\pi }{9}]]$

$=\sec [-{\sin }^{-1}\left[\sin (6\pi -\frac{4\pi }{9})\right]+{\cos }^{-1}\cos [3\pi +\frac{4\pi }{9}]]$

$=\sec [-{\sin }^{-1}[-\sin \frac{4\pi }{9}]+{\cos }^{-1}[-\cos \frac{4\pi }{9}]]$ .... $\left[\sin \right(6\pi -x)=-\sin x\text{and}\cos (3\pi +x)=-\cos x]$

$=\sec [{\sin }^{-1}\left[\sin \left(\frac{4\pi }{9}\right)\right]+\pi -{\cos }^{-1}\left[\cos \frac{4\pi }{9}\right]]$ ..... $[\because {\sin }^{-1}(-x)=-{\sin }^{-1}x\text{and}{\cos }^{-1}(-x)=\pi -{\cos }^{-1}x]$

$=\sec [\frac{4\pi }{9}+\pi -\frac{4\pi }{9}]=\sec \pi =-1$