Question

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

• A. $1$
• B. $-1$
• C. $2$
• D. $-4$

Answer 1

Answer: (C)

$2$

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

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

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

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

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

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

$=2$

Hence, the answer is $2.$