Question

If cosec $\theta$ - cot $\theta$=k, find the value of $\tan \frac{\theta }{2}$.

Answer 1

$cosec\theta -\cot \theta =k$

$\Rightarrow \frac{1-\cos \theta }{\sin \theta }=k$

$\Rightarrow \frac{2{\cos }^2\frac{\theta }{2}}{2\sin \frac{\theta }{2}\cos \frac{\theta }{2}}=k$

$\Rightarrow \frac{\cos \frac{\theta }{2}}{\sin \frac{\theta }{2}}=k$

$\Rightarrow \cot \frac{\theta }{2}=k$

$\Rightarrow \cot \frac{\theta }{2}=\frac{1}{k}$