Let $P(x)$ be a polynomial with real coefficients such that $P(\sin^2x) = P(\cos^2x)$ for all x in interval [0,π/2] Find which of the following statements are true ?
- $P(x)$ is an even function
- $P(x)$ can be expressed as a polynomial in $(2x - 1)^2 $
- $P(x)$ is a polynomial of even degree.
I tried taking a general poynomial and tried equating bpth sides. I am confused whether to replace x by π/2 - x or replace it by 1 - x while equating and how to move further. Or is there any other simpler approach ?