Question

lf $x=ta{n}^{-1}\left(1\right)+co{s}^{-1}\left(\frac{1}{2}\right)+si{n}^{-1}\left(\frac{1}{2}\right)$ and
$y=\cos \left[\frac{1}{2}co{s}^{-1}\right(1/8\left)\right]$ then

• A. $x=2\pi y$
• B. $y=3\pi x$
• C. $x=\pi y$
• D. $y=\pi x$

Answer 1

Answer: (C)

$x=\pi y$

If $x=ta{n}^{-1}\left(1\right)+co{s}^{-1}(-\frac{1}{2})+si{n}^{-1}(-\frac{1}{2})$
$=\frac{\pi }{4}+\frac{\pi }{2}[\because si{n}^{-1}x+co{s}^{-1}x=\frac{\pi }{2}]$
$x=\frac{3\pi }{4}$
$\therefore y=cos\left[\frac{1}{2}co{s}^{-1}\right(\frac{1}{8}\left)\right]cos\alpha =\frac{1}{8}$
$=cos\left(\frac{\alpha }{2}\right)[cos\alpha =2co{s}^2\frac{\alpha }{2}-1]$
$y=\sqrt{\frac{1+cos\alpha }{2}}=\sqrt{\frac{9}{16}}=\frac{3}{4}$
$\therefore x=\frac{3}{4}\pi =y\pi$