Q61 / 160 · AP EAMCET 2021 (25 Aug Shift 2) · Mathematics

If $0 < \theta < \frac{\pi}{2}$ and $\sin \theta \cos \theta=\frac{12}{25}$, then $\sin ^4 \theta+\cos ^4 \theta$ is equal to

  • A
    $\frac{327}{625}$
  • Correct answer
    B
    $\frac{337}{625}$
  • C
    $\frac{347}{625}$
  • D
    $\frac{340}{625}$

Solution

Given $0 < \theta < \frac{\pi}{2}$ and $\sin \theta \cdot \cos \theta=\frac{12}{25}$ To Find $\sin ^4 \theta+\cos ^4 \theta=$ ? $$ \begin{gathered} \sin ^4 \theta+\cos ^4 \theta=\left(\sin ^2 \theta\right)^2+\left(\cos ^2 \theta\right)^2 \\ \left\{\because a^2+b^2=(a+b)^2-2 a b\right\} \\ =\left(\sin ^2 \theta+\cos ^2 \theta\right)^2-2(\sin \theta \cdot \cos \theta)^2 \\ =(1)^2-2\left(\frac{12}{25}\right)^2=1-\frac{288}{625}=\frac{337}{625} \end{gathered} $$

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

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