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Trigonometry – II — Maths STD 11 Science — Question

Maharashtra BoardEnglish MediumSTD 11 ScienceMathsTrigonometry – II2 Marks
MCQ
Maximum and minimum values of $\sin ^4 \theta+\cos ^4 \theta$ are
  • A
    $0$,2
  • ✓
    $1, \frac{1}{2}$
  • C
    $-1,1$
  • D
    $1,-\frac{1}{2}$
✓

Answer

Correct option: B.
$1, \frac{1}{2}$
(B)
$\sin ^4 \theta+\cos ^4 \theta=\left(\sin ^2 \theta+\cos ^2 \theta\right)^2$$-2 \sin ^2 \theta \cos ^2 \theta$
$=1-\frac{1}{2}(\sin 2 \theta)^2$
Since $0 \leq \sin ^2 2 \theta \leq 1$
$\therefore \quad 0 \geq-\frac{1}{2} \sin ^2 2 \theta \geq-\frac{1}{2}$
$\Rightarrow 1+0>1-\frac{1}{2} \sin ^2 2 \theta>1-\frac{1}{2}$
$\Rightarrow 1 \geq \sin ^4 \theta+\cos ^4 \theta \geq \frac{1}{2}$

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