So I came across this question:
Find all possible values of $x$, $y$ satisfying:
$$\sin^2x + 4 \sin^2y – \sin x – 2\sin y – 2\sin x \sin y + 1 = 0 \qquad\forall\space x, y \in \left[0, \frac{\pi}{2}\right]$$
I guessed one way would be to write the expression to the left as the sum of three squares:
$$(p\sin x + q)^2 + (u\sin y + v)^2 + (l \sin x + m\sin y)^2 = 0$$
where $p$, $q$, $u$, $v$, $l$, $m$ are real numbers such that the respective coefficients of the term add up as they should. But it appears that those constants wouldn't be simple whole numbers or even numbers which I can get from trial and error methods.
So how do I go about solving such an equation?
Best Answer
Hint: With $$a=\sin(x),b=\sin(y)$$ we get $$a^2+4b^2-a-2b-2ab+1=0$$