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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

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

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