Answer:
\[y=\,\sin
\,\omega t-\,\cos \,\omega t\]
\[=\,\sin
\omega \,t+\,\sin \,\left( \omega t-\,\frac{\pi }{2} \right)\]
\[=2\,sin\,\left[
\frac{\omega t+\,\omega t-\,\frac{\pi }{2}}{2} \right]\,\,\cos \,\left[
\frac{\omega t\,-\omega t\,+\frac{\pi }{2}}{2} \right]\]
\[=2\,\sin
\,\left[ \omega t-\,\frac{\pi }{4} \right]\,\cos \left( \frac{\pi }{4}
\right)\]
\[=\,\frac{2}{\sqrt{2}}\,\sin
\,\left( \omega t-\frac{\pi }{4} \right)\,=\,\sqrt{2}\,\sin \,\left( \omega
t-\frac{\pi }{4} \right)\]
\[\therefore
\] Time period, \[T=\,\frac{2\pi
}{\omega }\].
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