Integral $n$ for $\sin^n(x) + \cos^n(x) = 1-\frac{n}{2}\sin^2(x)\cos^2(x)$

Question

Show that only integral solution of $\sin^n(x) + \cos^n(x) = 1-\frac{n}{2}\sin^2(x)\cos^{2}(x)$ is $n=4,6$.

I have proved for $n=4,6$ it is true, for other integer, I tried to check range of functions on left and right side. This is not as effective, and I failed. Please suggest a method in this.

Answer 1

$x=\pi/4$ gives $$ \frac{1}{2^{n/2-1}}= 1-\frac{n}{8} $$ and the RHS goes to $-\infty$ where LHS goes to $0$ as $n\to \infty$.

Answer 2

If you plug $x=\dfrac{\pi}{4}$, you will get:

$$\dfrac{2}{\sqrt{2}^n}=1-\dfrac{n}{8}.$$

Your left hand side is positive, so your right hand side should be as well, which imposes $n< 8$.

Now, if $n$ is odd, your left hand side is not rational, so you are left to check for $2,4,6$.

If $n=2$, the equality does not hold. Since you checked that $n=4,6$ work , we are done.

[Edit: $n$ replaced by $x$ on the first line]