# Trigonometric equation with 2 variables

Find all real values of $$x$$ and $$y$$ such that $$\sin^4x+\cos^4y+2=4\sin x\cos y$$ .

I started with $$u=\sin x$$, $$v=\cos y$$ I then can show that the above expression can be written as $$(u^2-1)+(v^2-1)+2 (u-v)^2=0$$ After this I am unable to find the values of $$u$$ and $$v$$. Please can anyone help me? Thank you

• It should be $\left(u^2-1\right)^2+\left(v^2-1\right)^2+2(u-v)^2=0$. – Lozenges Nov 18 '18 at 8:09

With the correction pointed out by @Lozenges, you have a sum of square numbers that is equal to $$0$$. That is true only if all terms are $$0$$. The solution is then given by $$\sin x=\cos y=\pm 1$$ You can write \begin{align}x&=\frac{\pi}2+n\pi\\y&=n\pi\end{align} with $$n\in\mathbb Z$$.