Question

If $cosec\frac{\theta }{2}-\cot \frac{\theta }{2}=\frac{1}{2},$ then the value of $cosec\frac{\theta }{2}+\cot \frac{\theta }{2}$ is

• A. 0
• B. 1
• C. 2
• D. $\frac{1}{2}$
  

Answer 1

The correct option is C 2

Given: $cosec\frac{\theta }{2}-\cot \frac{\theta }{2}=\frac{1}{2}$

Now, $1+{\cot }^2\theta =cose{c}^2\theta ,\text{for all}\theta$

$\Rightarrow 1=cose{c}^2\frac{\theta }{2}-{\cot }^2\frac{\theta }{2}$

$\Rightarrow 1=(cosec\frac{\theta }{2}+\cot \frac{\theta }{2})(cosec\frac{\theta }{2}-\cot \frac{\theta }{2})$

$\Rightarrow 1=(cosec\frac{\theta }{2}+\cot \frac{\theta }{2})\times \frac{1}{2}$ $(\because cosec\frac{\theta }{2}-\cot \frac{\theta }{2}=\frac{1}{2})$

$\Rightarrow cosec\frac{\theta }{2}+\cot \frac{\theta }{2}=2$

Hence, the correct answer is option (c).