Question

Solve :-
$(1+\tan \theta +\sec \theta )(1+\cot \theta -cosec\theta )=2$

Answer 1

$(1+\tan \theta +\sec \theta )(1+\cot \theta -cosec\theta )=2$

L.H.S.

$=(1+\cot \theta -cosec\theta )(1+\tan \theta +\sec \theta )$

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

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

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

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

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

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

$=2$

$=$R.H.S

$\therefore$ L.H.S$=$R.H.S

We proved.