Question

State whether the given equation is true or false:
$\frac{\cot \theta +cosec\theta -1}{\cot \theta -\mathrm{cosec}\theta +1}=\cot \frac{\theta }{2}$

• A. True
• B. False

Answer 1

The correct option is A True

$\frac{\cot \theta +\csc \theta -1}{\cot \theta -\csc \theta +1}$

$=\frac{\cot \theta +\csc \theta -({\csc }^2\theta -{\cot }^2\theta )}{\cot \theta -\csc \theta +1}$

$=\frac{\cot \theta +\csc \theta -(\csc \theta -\cot \theta )(\csc \theta +\cot \theta )}{\cot \theta -\csc \theta +1}$

$=\frac{(\csc \theta +\cot \theta )(1-\csc \theta +\cot \theta )}{\cot \theta -\csc \theta +1}$

$=\csc \theta +\cot \theta$

$=\frac{1}{\sin \theta }+\frac{\cos \theta }{\sin \theta }$

$=\frac{1+\cos \theta }{\sin \theta }$

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

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

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

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

Hence the given statement is true.