Question

The expression $f\left(\theta \right)=\frac{1}{\tan \theta +\sec \theta +\cot \theta +\text{cosec}\theta }$ is equivalent to

• A. $\frac{\text{cosec}\theta +\sec \theta +\text{cosec}\theta \sec \theta }{2\text{cosec}\theta \sec \theta }$
• B. $\frac{\text{cosec}\theta -\sec \theta -\text{cosec}\theta \sec \theta }{2\text{cosec}\theta \sec \theta }$
• C. $\frac{\text{cosec}\theta +\sec \theta -\text{cosec}\theta \sec \theta }{2\text{cosec}\theta \sec \theta }$
• D. None of these

Answer 1

Answer: (C)

$\frac{\text{cosec}\theta +\sec \theta -\text{cosec}\theta \sec \theta }{2\text{cosec}\theta \sec \theta }$

$f\left(\theta \right)=\frac{1}{\tan \theta +\sec \theta +\cot \theta +\text{cosec}\theta }$

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

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

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

$=\frac{\cos \theta +\sin \theta -1}{2}=\frac{\text{cosec}\theta +\sec \theta -\text{cosec}\theta .\sec \theta }{2\text{cosec}\theta .\sec \theta }$ .....(.dividing by $\cos \theta \sin \theta$)