$E(x)=\sqrt{\cos^{4}(x)+m\cdot\sin^{2}(x)}+\sqrt{\sin^{4}(x)+m\cdot\cos^{2}(x)}$ is constant for $m=?$

Question

The values of $m$ such that $E(x)$ is constant.

$$E(x)=\sqrt{\cos^{4}(x)+m\cdot\sin^{2}(x)}+\sqrt{\sin^{4}(x)+m\cdot\cos^{2}(x)}$$

I tried to take this expression as a function then to find it's derivative and to equalize it with 0 but I didn't get too far.Also, I tried squaring but I got heavy calculations.Some ideas?

Answer 1

As $E(x)$ is a constant, $E(0)=E(\pi/4)$.

\begin{align*} 1+\sqrt{m}&=\sqrt{\frac{1}{4}+\frac{m}{2}}+\sqrt{\frac{1}{4}+\frac{m}{2}}\\ 1+2\sqrt{m}+m&=1+2m\\ 2\sqrt{m}&=m\\ m&=0 \quad\textrm{or}\quad 4 \end{align*}

When $m=0$, $E(x)=1$ for all $x$.

When $m=4$,

\begin{align*} E(x)&=\sqrt{\cos^{4}x+4\sin^{2}x}+\sqrt{\sin^{4}x+4\cos^{2}x}\\ &=\sqrt{(1-\sin^2x)^2+4\sin^2x}+\sqrt{(1-\cos^2x)^2+4\cos^2x}\\ &=(1+\sin^2x)+(1+\cos^2x)\\ &=3 \end{align*}