We know that ( \cos(3\pi/2 - x) = \sin(x) ). Therefore, ( \cos^2(3\pi/2 - x) = \sin^2(x) ).
Substitute ( \cos^2(3\pi/2 - x) ) with ( \sin^2(x) ).
So, the equation becomes ( 2 \sin^2(x) = -\sin(x) ).
Divide by ( \sin(x) ) on both sides.
( 2 \sin(x) = -1 ).
This cannot be true since the left side can only range from -2 to 2, while the right side is -1. Therefore, there is no solution for the given equation.
Let's simplify the given equation step by step.
We know that ( \cos(3\pi/2 - x) = \sin(x) ). Therefore, ( \cos^2(3\pi/2 - x) = \sin^2(x) ).
Substitute ( \cos^2(3\pi/2 - x) ) with ( \sin^2(x) ).
So, the equation becomes ( 2 \sin^2(x) = -\sin(x) ).
Divide by ( \sin(x) ) on both sides.
( 2 \sin(x) = -1 ).
This cannot be true since the left side can only range from -2 to 2, while the right side is -1. Therefore, there is no solution for the given equation.