Q62 / 160 · TS EAMCET 2009 · Mathematics

The period of $\sin ^4 x+\cos ^4 x$ is

  • A
    $\frac{\pi^4}{2}$
  • B
    $\frac{\pi^2}{2}$
  • C
    $\frac{\pi}{4}$
  • Correct answer
    D
    $\frac{\pi}{2}$

Solution

Let $\begin{aligned} f(x) & =\sin ^4 x+\cos ^4 x \\ & =\left(\sin ^2 x+\cos ^2 x\right)^2-2 \sin ^2 x \cos ^2 x \\ & =1-\frac{1}{4} \cdot 2(\sin 2 x)^2\end{aligned}$ $\begin{aligned} & =1-\frac{1}{4}(1-\cos 4 x) \\ & =\frac{3}{4}+\frac{\cos 4 x}{4} \\ \therefore \text { Period of } f(x) & =\frac{2 \pi}{4}=\frac{\pi}{2}\end{aligned}$

Hence, the correct answer is option: 'D'.

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