Question

Simplify $(cosec\theta +cot\theta )\times (1-cos\theta ).$

• A. $cot\theta$
• B. $sec\theta$
• C. $sin\theta$
• D. $cosec\theta$

Answer 1

The correct option is C $sin\theta$

We know that,
$cosec\theta =\frac{1}{sin\theta },cot\theta =\frac{cos\theta }{sin\theta }$

On substituting these values,
$(cosec\theta +cot\theta )\times (1-cos\theta )=(\frac{1}{sin\theta }+\frac{cos\theta }{sin\theta })\times (1-cos\theta )$

On multiply $\frac{(1+cos\theta )}{(1+cos\theta })$, we get

$(\frac{1}{sin\theta }+\frac{cos\theta }{sin\theta })\times \frac{(1-cos\theta )\times (1+cos\theta )}{(1+cos\theta )}=\frac{(1+cos\theta )}{sin\theta }\times \frac{(1-cos\theta )\times (1+cos\theta )}{(1+cos\theta )}=\frac{(1+cos\theta )}{sin\theta }\times \frac{(1-co{s}^2\theta )}{(1+cos\theta )}$
By cancelling out the common terms,
$=\frac{(1-co{s}^2\theta )}{sin\theta }$

$\because$ $1-co{s}^2\theta =si{n}^2\theta$, on further simplifying,
$\Rightarrow \frac{(1-co{s}^2\theta )}{sin\theta }=\frac{si{n}^2\theta }{sin\theta }=sin\theta$

$\therefore (cosec\theta +cot\theta )\times (1-cos\theta )=sin\theta$