Let sin x + sin y = cos x + cos y for all x, y ∈ ℝ. What is $\rm tan \left(\frac{x}{2}+\frac{y}{2}\right)$ equal to?

Let $A = \begin{bmatrix} 0 & 2 \\ -2 & 0 \end{bmatrix}$ and (mI + nA)² = A where m, n are positive real numbers and I is the identify matrix. What is (m + n) equal to?

What is the value of following?

$\rm cot \left[sin^{-1} \frac{3}{5}+cot^{-1}\frac{3}{2} \right]$

Let 4sin² x = 3, where 0 ≤ x ≤ π. What is tan3x equal to?