The displacement of an oscillator is given by $x = a\, \sin \, \omega t + b\, \cos \, \omega t$. where $a, b$ and $\omega$ are constant. Then :-
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$x=\sin \omega t+b \cos \omega t$
$\mathrm{x}=\sqrt{\mathrm{a}^{2}+\mathrm{b}^{2}}\left[\frac{\mathrm{a}}{\mathrm{a}^{2}+\mathrm{b}^{2}} \sin \omega \mathrm{t}+\frac{1}{\sqrt{\mathrm{a}^{2}+\mathrm{b}^{2}}} \cos \omega \mathrm{t}\right]$
$\mathrm{x}=\sqrt{\mathrm{a}^{2}+\mathrm{b}^{2}}[\cos \phi \sin \omega \mathrm{t}+\sin \phi \cos \omega \mathrm{t}]$
Let $\cos \phi=\frac{a}{\sqrt{a^{2}+b^{2}}}$
$\therefore \mathrm{x}_{2} \sqrt{\mathrm{a}^{2}+\mathrm{b}^{2}} \sin (\omega \mathrm{t}+\phi)$ this is condition of $SHM$
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