Question

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

Answer 1

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

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

$=\frac{\cos \theta +1}{\sin \theta }=\frac{\cos \theta }{\sin \theta }+\frac{1}{\sin \theta }=co\sec \theta +\cot \theta$ = RHS

$\therefore$ LHS = RHS