$sin²3x + sin²4x = sin²5x + sin²6x$
$⇔$ $1 - cos²3x + 1 - cos²4x = 1 - cos²5x + 1 - cos²6x$
$⇔$ $cos²3x + cos²4x = cos²5x + cos²6x$
$⇔$ $1 + cos6x + 1 + cos8x = 1 + cos10x + 1 + cos12x$
$⇔$ $cos6x + cos8x = cos10x + cos12x$
$⇔$ $2.cosx.cos7x = 2.cosx.cos11x$
$⇔$ $cosx.cos7x - cosx.cos11x = 0$
$⇔$ $cosx.( cos 7x - cos11x ) = 0$
$⇔$ $\left[ \begin{array}{l}cosx=0\\cos7x = cos11x\end{array} \right.$
Với $cosx = 0 ⇔ x = \frac{\pi}{2} + k\pi$
Với $cos7x = cos11x ⇔ \left[ \begin{array}{l}7x = 11x + k2\pi\\7x = -11x + k2\pi\end{array} \right.$
$⇔ \left[ \begin{array}{l}x=k\frac{\pi}{2}\\x=k\frac{\pi}{9}\end{array} \right.$
