Question

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

• A. $0$
• B. $1$
• C. $2$
• D. $-1$

Answer 1

Answer: (C)

$2$

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

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

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

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

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