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  4. The period of the function f(x)= sin 4x+ cos 4x is:

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Q. The period of the function $ f(x)={{\sin }^{4}}x+{{\cos }^{4}}x $ is:

JamiaJamia 2005

Solution:

$ f(x)={{\sin }^{4}}x+{{\cos }^{4}}x $ $ ={{({{\sin }^{2}}x+{{\cos }^{2}}x)}^{2}}-2{{\sin }^{2}}x{{\cos }^{2}}x $ $ =1-\frac{1}{2}{{(\sin 2x)}^{2}}=1-\frac{1}{2}\left[ \frac{1-\cos 4x}{2} \right] $ $ =\frac{3}{4}+\frac{1}{4}\cos 4x $ Since, $ \cos x $ is periodic with period $ 2\pi $ $ \therefore $ The period off $ f(x)=\frac{2\pi }{4}=\frac{\pi }{2} $