Question

Prove that $\frac{\cos \theta -\sin \theta +1}{\cos \theta +\sin \theta -1}=\csc \theta +\cot \theta$

Answer 1

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

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

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

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

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

$=\frac{2{\cos }^2\theta +2\cos \theta }{2\sin \theta \cos \theta }=\text{cosec}\theta +\cot \theta =RHS$

Hence, Proved