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Trigonometric Functions — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsTrigonometric Functions2 Marks
MCQ
$\cot \theta=\sin 2 \theta(\theta \neq n \pi, n$ is an integer $)$, if $\theta=$
  • A
    $45^{\circ}$ and $60^{\circ}$
  • ✓
    $45^{\circ}$ and $90^{\circ}$
  • C
    $45^{\circ}$ only
  • D
    $90^{\circ}$ only
✓

Answer

Correct option: B.
$45^{\circ}$ and $90^{\circ}$
(B) $\cot \theta=\sin 2 \theta,(\theta \neq n \pi)$
$\Rightarrow \frac{\cos \theta}{\sin \theta}=2 \sin \theta \cos \theta$
$\begin{array}{l}\Rightarrow 2 \sin ^2 \theta \cos \theta=\cos \theta \\ \Rightarrow \cos \theta\left(2 \sin ^2 \theta-1\right)=0\end{array}$
$\Rightarrow \cos \theta=0$ or $\sin ^2 \theta=\frac{1}{2}$
$\Rightarrow \cos \theta=0$ or $\sin ^2 \theta=\sin ^2\left(\frac{\pi}{4}\right)$
$\Rightarrow \theta=(2 n+1) \frac{\pi}{2}$ or $\theta=n \pi \pm \frac{\pi}{4}$
$\Rightarrow \theta=90^{\circ}$ and $45^{\circ}$ $\ldots\left[\begin{array}{l}\because \text { at } \theta=90^{\circ} \text { and } 45^{\circ}, \\ \text { the given equation is satisfied. }\end{array}\right]$

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