Question

$(\text{cosec}\theta -\sin \theta )(\sec \theta -\cos \theta )(\tan \theta +\cot \theta )=.......$

• A. $0$
• B. $1$
• C. $-1$
• D. None of above

Answer 1

Answer: (B)

$1$

$(\text{cosec}\theta -\sin \theta )(\sec \theta -\cos \theta )(\tan \theta +\cot \theta )$

$=(\frac{1}{\sin \theta }-\sin \theta )(\frac{1}{\cos \theta }-\cos \theta )(\frac{\sin \theta }{\cos \theta }+\frac{\cos \theta }{\sin \theta })$, using basic identities
$=\left(\frac{1-{\sin }^2\theta }{\sin \theta }\right)\left(\frac{1-{\cos }^2\theta }{\cos \theta }\right)\left(\frac{{\sin }^2\theta +{\cos }^2\theta }{\sin \theta \cdot \cos \theta }\right)$
$=\frac{{\cos }^2\theta }{\sin \theta }\times \frac{{\sin }^2\theta }{\cos \theta }\times \frac{1}{\sin \theta \cdot \cos \theta }=1\times 1\times 1=1$