Question

Let $f\left(x\right)={\sin }^{23}x-{\cos }^{22}x$ and $g\left(x\right)=1+\frac{1}{2}{\tan }^{-1}\left|x\right|.$
Then, the number of values of $x$ in interval $[-10\pi ,20\pi ]$ satisfying the equation $f\left(x\right)=sgn\left(g\right(x\left)\right),$ is $5a$ Then, $a$ is

Answer 1

$g\left(x\right)=\frac{1}{2}{\tan }^{-1}\left|x\right|+1$
$\Rightarrow sgn\left(g\right(x\left)\right)=1$
${\sin }^{23}x-{\cos }^{22}x=1$
${\sin }^{23}x=1+{\cos }^{22}x$
which is possible ,if $\sin x=1$ and $\cos x=0\Rightarrow \sin x=1,x=2n\pi +\frac{\pi }{2}$
Hence, $-10\pi \le 2n\pi +\frac{\pi }{2}\le 20\pi$
$\Rightarrow -\frac{21}{2}\le 2n\le \frac{39}{2}$
$\Rightarrow -\frac{21}{4}\le n\le \frac{39}{4}\Rightarrow -5\le n\le 9$
Hence, number of values of $x=15\Rightarrow a=3$
Ans: 3