Question

Choose the correct option. Justify your choice.
$(1+\tan \theta +\sec \theta )(1+\cot \theta -cosec\theta )=?$

• A. $0$
• B. $1$
• C. $2$
• D. $-1$

Answer 1

The correct option is C $2$

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

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

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

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

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

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

$=2$

So, $(1+\tan \theta +\sec \theta )(1+\cot \theta -\csc \theta )=2$.