Explanation

(c) The given equation is written as
$y = a(\sin{\omega t + \cos{\omega t)}}\ $
or $y = a\sqrt{2}\left( {\frac{1}{\sqrt{2}}\sin}{\omega t + \frac{1}{\sqrt{2}}\cos{\omega t}} \right)\ $
or $y = a\sqrt{2}\left\lbrack \cos\frac{\pi}{4}\sin{\omega t + \sin\frac{\pi}{4}\cos{\omega t}} \right\rbrack\ $
or $y = a\sqrt{2}\ \sin{(\omega t +}\frac{\pi}{4})\ $
Thus, we have seen that the particle’s motion is simple harmonic with amplitude $a\sqrt{2}$.