Question

Prove the following identity:
$\frac{\tan \theta }{\sec \theta +1}+\frac{\cot \theta }{cosec\theta +1}=cosec\theta +\sec \theta -\sec \theta .cosec\theta$

Answer 1

Now,

$\frac{\tan \theta }{\sec \theta +1}+\frac{\cot \theta }{cosec\theta +1}$

$=\frac{\tan \theta (\sec \theta -1)}{{\sec }^2\theta -1}+\frac{\cot \theta (cosec\theta -1)}{cose{c}^2\theta -1}$

$=\frac{(\sec \theta -1)}{\tan \theta }+\frac{(cosec\theta -1)}{\cot \theta }$

$=cosec\theta -\cot \theta +\sec \theta -\tan \theta$

$=cosec\theta +\sec \theta -\tan \theta -\cot \theta$

$=cosec\theta +\sec \theta -\left(\frac{{\cot }^2\theta +1}{\cot \theta }\right)$

$=cosec\theta +\sec \theta -\left(\frac{cose{c}^2\theta }{\cot \theta }\right)$

$=cosec\theta +\sec \theta -cosec\theta .\sec \theta$