Solve $[\sin(\frac{\pi}{5})]^{x}+[\cos(\frac{\pi}{5})]^x$=1 Solve $[\sin(\frac{\pi}{5})]^{x}+[\cos(\frac{\pi}{5})]^x=1$

Here is another problem that I don't know how to solve. I know that $\sin^2(x)+\cos^2(x)=1$. How do I prove that here $x=2$?. Isn't this just a particular case of proving $\sin^2(kx)+\cos^2(kx)=1$ with $k$ being an integer?
Because both $sin\left(\frac\pi5\right)$ and $cos\left(\frac\pi5\right)$ are fractions, each will become smaller as you raise them to successively higher powers.  At $x=2$ they already add to exactly $1$.  Adding two smaller numbers will never add up to $1$.
http://www.wolframalpha.com/input/?i=sin+(pi%2F5)%5Ex+%2B+cos+(pi%2F5)%5Ex+%3D+1
This might help by equating LHS to RHS.

Here $(\frac {5}{8}-\frac {\sqrt 5}{8})^{\frac x2}+(\frac 14(1+\sqrt5))^x=1$.
Simplify and solve equation you will get x=2.
Also plotting graph will give the rational solution.