Question

The number of real roots of the equation ${\sin }^{2020}x-{\cos }^{2020}x+2019=2020$ in the interval $\left(-\frac{3\pi }{2},\frac{5\pi }{2}\right)$

• A. 1
• B. 3
• C. 5
• D. infinitely many

Answer 1

Answer: (B)

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 $x=-\frac{\pi }{2},\frac{\pi }{2},\frac{3\pi }{2}$