Find the period of f(x)=sin^4 x +cos^4 x

Solution:      We have ,    $f(x)=\sin^{4}x+\cos^{4}x$

$\\ \\ \Rightarrow \hspace{1cm}f(x)=\sin^{4}x+\cos^{4}x=(\sin^{2}x+\cos^{2}x)^{2}-2\sin^{2}\cos^{2}x\\ \\ \Rightarrow \hspace{1cm}f(x)=1-\frac{1}{2}\sin^{2}2x=1-\frac{1}{4}(1-\cos 4x)\\ \\ \Rightarrow \hspace{1cm}f(x)=\frac{3}{4}+\frac{1}{4}\sin (4x+\frac{\pi}{2}).$

Whence ,              $T=\frac{2\pi}{\omega}=\frac{2\pi}{4}=\frac{\pi}{2}.$

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