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Q.
The number of real roots of the equation $\sin ^{2020} x-\cos ^{2020} x+2019=2020$ in the interval $\left(-\frac{3 \pi}{2}, \frac{5 \pi}{2}\right)$
TS EAMCET 2020
Solution:
Given equation
$\sin ^{2020} x-\cos ^{2020} x+2019=2020$
$\Rightarrow \sin ^{2020} x=1+\cos ^{2020} x$
the range of $LHS$ is $[0,1]$ and
the range of RHS is $[1,2]$
So, for the solution $\sin ^{2020} x=1$ and $\cos ^{2020} x=0$
and in the given internal $\left(-\frac{3 \pi}{2}, \frac{5 \pi}{2}\right)$,
the possible values of $x=-\frac{\pi}{2}, \frac{\pi}{2}, \frac{3 \pi}{2}$