Question

Prove it :
$\sqrt{\frac{1+\sin \theta }{1-\sin \theta }}=\frac{1+\cot \frac{\theta }{2}}{1-\cot \frac{\theta }{2}}$

Answer 1

$LHS=\sqrt{\frac{1+\sin \theta }{1-\sin \theta }}$

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

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

$=\sqrt{{\left(\frac{\cos \frac{\theta }{2}+\sin \frac{\theta }{2}}{\cos \frac{\theta }{2}-\sin \frac{\theta }{2}}\right)}^2}$

$=\left|\frac{\cos \frac{\theta }{2}+\cos \frac{\theta }{2}}{\cos \frac{\theta }{2}-\sin \frac{\theta }{2}}\right|$

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

$=\frac{1+\cot \frac{\theta }{2}}{1-\cot \frac{\theta }{2}}=RHS$