Question

$(1+\cot \theta -\text{cosec}\theta )(1+\tan \theta +\sec \theta )$ equals:

• A. $1$
• B. $2$
• C. $\cot \theta$
• D. $(1+\cot \theta )(1+\tan \theta )$

Answer 1

The correct option is B $2$

Given: $(1+\cot \theta -\text{cosec}\theta )(1+\tan \theta +\sec \theta )$

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

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

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

$={\sin }^2\theta +{\cos }^2\theta +2\sin \theta \cos \theta -1$ {$si{n}^{2}A+co{s}^{2}A=1$}

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

$=2$

Hence, option B is the correct option.