Solving a trigonometric equation

Question

Can someone help me to solve this problem?

Find all number pairs $x,y$ that satisfy the equation:

$$\tan^4(x) + \tan^4(y) + 2\cot^2(x)\cot^2(y) = 3 + \sin^2(x+y)$$

Answer 1

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.