Question

Find the value of $(1+\cot \theta -\text{cosec}\theta )(1+\tan \theta +\sec \theta )$

• A. $1$
• B. $2$
• C. $0$
• D. $-1$

Answer 1

The correct option is B $2$

$(1+\cot \theta -\csc \theta )(1+\tan \theta +\sec \theta )$

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

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

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

$=\left[\frac{{\sin }^2\theta +{\cos }^2\theta +2\sin \theta \cos \theta -1}{\sin \theta \cos \theta }\right]$

$=\left[\frac{1+2\sin \theta \cos \theta -1}{\sin \theta \cos \theta }\right]$

$=\left[\frac{2\sin \theta \cos \theta }{\sin \theta \cos \theta }\right]$

$=2$(Ans)