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Trigonometry – II — Maths STD 11 Science — Question

Maharashtra BoardEnglish MediumSTD 11 ScienceMathsTrigonometry – II2 Marks
MCQ
$2 \sin ^2 \beta+4 \cos (\alpha+\beta) \sin \alpha \sin \beta +\cos 2(\alpha+\beta)=$
  • A
    $\sin 2 \alpha$
  • B
    $\cos 2 \beta$
  • ✓
    $\cos 2 \alpha$
  • D
    $\sin 2 \beta$
✓

Answer

Correct option: C.
$\cos 2 \alpha$
(C)
$\cos 2(\alpha+\beta)=2 \cos ^2(\alpha+\beta)-1$
and $2 \sin ^2 \beta=1-\cos 2 \beta$
Now, $2 \sin ^2 \beta+4 \cos (\alpha+\beta) \sin \alpha \sin \beta$$+\cos 2(\alpha+\beta)$
$=1-\cos 2 \beta+4 \cos (\alpha+\beta) \sin \alpha \sin \beta$$+2 \cos ^2(\alpha+\beta)-1$
$=2 \cos (\alpha+\beta)[2 \sin \alpha \sin \beta$$+\cos (\alpha+\beta)]-\cos 2 \beta$
$=-\cos 2 \beta+2 \cos (\alpha+\beta) \cos (\alpha-\beta)$
$=-\cos 2 \beta+\cos 2 \alpha+\cos 2 \beta$
$=\cos 2 \alpha$

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