MCQ
If $\sin\theta=\frac{\sqrt3}{2}$ then $(\text{cosec }\theta+\cot\theta)=?$
- A$\big(2+\sqrt3\big)$
- B$2\sqrt3$
- C$\sqrt2$
- ✓$\sqrt3$
✓
Answer
Correct option: D.
$\sqrt3$
Given $\sin\theta=\frac{\sqrt3}{2}$ and $\text{cosec }\theta=\frac{2}{\sqrt3}$
$\text{cosec}^2\theta-\cot^2\theta=1$
$\Rightarrow\cot^2\theta=\text{cosec}^2\theta-1$
$\Rightarrow\cot^2\theta=\frac{4}3{}-1\ [$Given$]$
$\Rightarrow\cot\theta=\frac{1}{\sqrt3}$
$\therefore\text{cosec }\theta+\cot\theta=\frac{2}{\sqrt3}+\frac{1}{\sqrt3}$
$=\frac{2}{\sqrt3}=\frac{\sqrt3\times\sqrt3}{\sqrt3}=\sqrt3$
$\text{cosec}^2\theta-\cot^2\theta=1$
$\Rightarrow\cot^2\theta=\text{cosec}^2\theta-1$
$\Rightarrow\cot^2\theta=\frac{4}3{}-1\ [$Given$]$
$\Rightarrow\cot\theta=\frac{1}{\sqrt3}$
$\therefore\text{cosec }\theta+\cot\theta=\frac{2}{\sqrt3}+\frac{1}{\sqrt3}$
$=\frac{2}{\sqrt3}=\frac{\sqrt3\times\sqrt3}{\sqrt3}=\sqrt3$
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