$a) \, \sin2x +\cos2x = \sqrt2$
$\Leftrightarrow \sqrt2\sin\left(x + \dfrac{\pi}{4}\right) =\sqrt2$
$\Leftrightarrow \sin\left(x + \dfrac{\pi}{4}\right) =1$
$\Leftrightarrow x + \dfrac{\pi}{4} = \dfrac{\pi}{2} + k2\pi$
$\Leftrightarrow x = \dfrac{\pi}{4} + k2\pi \quad (k\in \Bbb Z)$
$b) \, \cos2x.\cos5x = \cos7x$
$\Leftrightarrow \dfrac{1}{2}(\cos7x + \cos3x) = \cos7x$
$\Leftrightarrow \cos7x = \cos3x$
$\Leftrightarrow \left[\begin{array}{l}7x = 3x +k2\pi\\7x = -3x + k2\pi\end{array}\right.$
$\Leftrightarrow \left[\begin{array}{l}x = k\dfrac{\pi}{2}\\x = k\dfrac{\pi}{5}\end{array}\right.\quad (k \in \Bbb Z)$
