Question

Prove that $(1+cot\Theta -cosec\Theta )(1+tan\Theta +sec\Theta )=2$.

Answer 1

$\left(1+\cot \theta -\cos ec\theta \right)\left(1+\tan \theta +\sec \theta \right)=2$

$L.H.S$ $\left(1+\cot \theta -\cos ec\theta \right)\left(1+\tan \theta +\sec \theta \right)$

$\begin{array}{c}=\left(1+\frac{\cos \theta }{\sin \theta }-\frac{1}{\sin \theta }\right)\left(1+\frac{\sin \theta }{\cos \theta }+\frac{1}{\cos \theta }\right)\\ =\left(\frac{\sin \theta +\cos \theta -1}{\sin \theta }\right)\left(\frac{\sin \theta +\cos \theta +1}{\cos \theta }\right)\\ =\frac{{\left(\sin \theta +\cos \theta \right)}^2-1}{\sin \theta \cos \theta }\\ =\frac{2\sin \theta \cos \theta +{\sin }^2\theta +{\cos }^2\theta -1}{\sin \theta \cos \theta }\\ =\frac{1+2\sin \theta \cos \theta -1}{\sin \theta \cos \theta }\\ =\frac{2\sin \theta \cos \theta }{\sin \theta \cos \theta }\\ =2\\ LHS=RHS\end{array}$