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Trigonometry – II — Maths STD 11 Science — Question

Maharashtra BoardEnglish MediumSTD 11 ScienceMathsTrigonometry – II2 Marks
MCQ
If $\sin 2 \theta+\sin 2 \phi=\frac{1}{2}$ and
$\cos 2 \theta+\cos 2 \phi=\frac{3}{2}$, then $\cos ^2(\theta-\phi)=$
  • A
    $\frac{3}{8}$
  • ✓
    $\frac{5}{8}$
  • C
    $\frac{3}{4}$
  • D
    $\frac{5}{4}$
✓

Answer

Correct option: B.
$\frac{5}{8}$
(B)
$\sin 2 \theta+\sin 2 \phi=\frac{1}{2}$
and $\cos 2 \theta+\cos 2 \phi=\frac{3}{2}$
Squaring and adding (i) and (ii), we get
$\left(\sin ^2 2 \theta+\cos ^2 2 \theta\right)+\left(\sin ^2 2 \phi+\cos ^2 2 \phi\right)$
$+2(\sin 2 \theta \sin 2 \phi+\cos 2 \theta \cos 2 \phi)=\frac{1}{4}+\frac{9}{4}$
$\Rightarrow \cos 2 \theta \cos 2 \phi+\sin 2 \theta \sin 2 \phi=\frac{1}{4}$
$\Rightarrow \cos (2 \theta-2 \phi)=\frac{1}{4}$
$\Rightarrow 2 \cos ^2(\theta-\phi)-1=\frac{1}{4}$
$\Rightarrow \cos ^2(\theta-\phi)=\frac{5}{8}$

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