Solving a trigonometric equation Can someone help me to solve this problem?
Find all number pairs $x,y$ that satisfy the equation: 
$$\tan^4(x) + \tan^4(y) + 2\cot^2(x)\cot^2(y) = 3 + \sin^2(x+y)$$

By AM-GM inequality:
$$(\tan x)^4 + (\tan y)^4 + (\cot x)^2\cdot (\cot y)^2 + (\cot x)^2\cdot (\cot y)^2 \geq 4 \geq 3 + (\sin(x+y))^2
$$
So the equation occurs when: $\sin(x+y) = 1, -1$, and $\tan x = \tan y, -\tan y$, and $\tan x = \cot x, -\cot x$. So $\tan x = 1, -1 = \tan y$. You can look at cases.
