Question

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

• A. $0$
• B. $1$
• C. $2$
• D. $-1$

Answer 1

The correct option is C $2$

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

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

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

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

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

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

$=2$