Question

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

• A. $\sec \frac{10\pi }{9}$
• B. $\sec \frac{\pi }{9}$
• C. $1$
• D. $-1$

Answer 1

The correct option is D $-1$

$sec\left[si{n}^{-1}\right(sin(-\frac{50\pi }{9}))+co{s}^{-1}(cos\left(\frac{-31\pi }{9}\right)\left)\right]$
$=sec[-si{n}^{-1}(sin\left(\frac{50\pi }{9}\right))+co{s}^{-1}(cos\left(\frac{31\pi }{9}\right)\left)\right]$
$=sec[-si{n}^{-1}(sin(6\pi -\frac{4\pi }{9}))+co{s}^{-1}(cos(3\pi +\frac{4\pi }{9})\left)\right]$
$=sec[-si{n}^{-1}(-sin\left(\frac{4\pi }{9}\right))+co{s}^{-1}(-cos\left(\frac{4\pi }{9}\right)\left)\right]$
$=sec\left[si{n}^{-1}\right(sin\left(\frac{4\pi }{9}\right))+\pi -co{s}^{-1}(cos\left(\frac{4\pi }{9}\right)\left)\right]$
$=sec[\frac{4\pi }{9}-\frac{4\pi }{9}+\pi ]$
$=sec\left[\pi \right]$
$=-1$