Question

Find if $\frac{\sin \theta -\cos \theta +1}{\sin \theta +\cos \theta +1}=\frac{1}{\sec \theta -\tan \theta }$ using the identity ${\sec }^2\theta =1+{\tan }^2\theta$

Answer 1

$LHS=\frac{\sin \theta -\cos \theta +1}{\sin \theta +\cos \theta +1}=\frac{2\sin \frac{\theta }{2}(\cos \frac{\theta }{2}+\sin \frac{\theta }{2})}{2\cos \frac{\theta }{2}(\sin \frac{\theta }{2}+\cos \frac{\theta }{2})}=\tan \frac{\theta }{2}$

$RHS=\frac{1}{\text{sec}\theta -\tan \theta }=\frac{\cos \theta }{1-\sin \theta }=\frac{2\sin (\frac{\pi }{4}+\frac{\theta }{2})\cos (\frac{\pi }{4}+\frac{\theta }{2})}{2{\cos }^2(\frac{\pi }{4}+\frac{\theta }{2})}=\tan (\frac{\pi }{4}+\frac{\theta }{2})$

$LHS\ne RHS$

SO given question is wrong