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Trigonometry – II — Maths STD 11 Science — Question

Maharashtra BoardEnglish MediumSTD 11 ScienceMathsTrigonometry – II2 Marks
MCQ
$\cos ^2 \alpha+\cos ^2\left(\alpha+120^{\circ}\right)+\cos ^2\left(\alpha-120^{\circ}\right)$ is equal to
  • ✓
    $\frac{3}{2}$
  • B
    1
  • C
    $\frac{1}{2}$
  • D
    $0$
✓

Answer

Correct option: A.
$\frac{3}{2}$
(A)
$\cos ^2 \alpha+\cos ^2\left(\alpha+120^{\circ}\right)+\cos ^2\left(\alpha-120^{\circ}\right)$
$=\cos ^2 \alpha+\left\{\cos \left(\alpha+120^{\circ}\right)+\cos \left(\alpha-120^{\circ}\right)\right\}^2$$-2 \cos \left(\alpha+120^{\circ}\right) \cos \left(\alpha-120^{\circ}\right)$
$=\cos ^2 \alpha+\left\{2 \cos \alpha \cos 120^{\circ}\right\}^2$$-2\left\{\cos ^2 \alpha-\sin ^2 120^{\circ}\right\}$
$=\cos ^2 \alpha+\cos ^2 \alpha-2 \cos ^2 \alpha+2 \sin ^2 120^{\circ}$
$=2 \sin ^2 120^{\circ}$
$=2 \times \frac{3}{4}=\frac{3}{2}$

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