$y=y_{1}+y_{2}$

$y=A \sin (w t)+A \cos (w t)$

$y=\sqrt{2} A\left[\frac{1}{\sqrt{2}} \times \sin (w t)+\frac{1}{\sqrt{2}} \times \cos (w t)\right]$

$y=\sqrt{2} A\left[\cos \left(\frac{\pi}{4}\right) \sin (w t)+\sin \left(\frac{\pi}{4}\right) \cos (w t)\right] \quad \Longrightarrow \quad y=\sqrt{2} A[\sin (w t+\pi / 4)]$

But for $SHM$ $y=B \sin (w t+\phi), \quad$ total mechanical energy $\quad T \cdot E=\frac{1}{2} m w^{2} B^{2}$

Thus total mechanical energy for $y, \quad T . E=\frac{1}{2} m w^{2}(\sqrt{2} A)^{2}$

$\Longrightarrow \quad T . E=m w^{2} A^{2}$