Question

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

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

Answer 1

The correct option is C

$\sin \theta$

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

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

On plugging the values of $cosec\theta$ and $\cot \theta$ and rationalising $(1-\cos \theta )$, we get

$(\frac{1}{\sin \theta }+\frac{\cos \theta }{\sin \theta })\times \frac{(1-\cos \theta )\times (1+\cos \theta )}{(1+\cos \theta )}$

$=\frac{(1+\cos \theta )}{\sin \theta }\times \frac{(1-{\cos }^2\theta )}{(1+\cos \theta )}$

$=\frac{1-{\cos }^2\theta }{\sin \theta }$

$\because {\sin }^2\theta +{\cos }^2\theta =1$
$\Rightarrow \frac{1-{\cos }^2\theta }{\sin \theta }=\frac{{\sin }^2\theta }{\sin \theta }$
$=\sin \theta$