Question

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

Solution

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

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