Question

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

Answer 1

$\left(1+\tan \theta +\sec \theta \right)\left(1+\cot \theta -\cos ec\theta \right)\Rightarrow \left(1+\frac{\sin \theta }{\cos \theta }+\frac{1}{\cos \theta }\right)\left(1+\frac{\cos \theta }{\sin \theta }-\frac{1}{\sin \theta }\right)\Rightarrow \left(\frac{\cos \theta +\sin \theta +1}{\cos \theta }\right)\left(\frac{\sin \theta +\cos \theta -1}{\sin \theta }\right)\Rightarrow \frac{{\left(\sin \theta +\cos \theta \right)}^2-1}{\sin \theta .\cos \theta }\left[\because {\sin }^2\theta +{\cos }^2\theta =1\right]\Rightarrow \frac{{\sin }^2\theta +{\cos }^2\theta +2\sin \theta .\cos \theta -1}{\sin \theta .\cos \theta }\Rightarrow \frac{\left(1-1\right)+2\sin \theta .\cos \theta }{\sin \theta .\cos \theta }=\frac{2\sin \theta .\cos \theta }{\sin \theta .\cos \theta }=2$