Question

Simplify the expression:
$\frac{1}{sin\theta +cos\theta }+\frac{1}{sin\theta -cos\theta }$

• A. 0
• B. $\frac{2.cos\theta }{1-2co{s}^2\theta }$
• C. $\frac{2.sin\theta }{1-2co{s}^2\theta }$
• D. $\frac{2.sin\theta }{1-2si{n}^2\theta }$

Answer 1

The correct option is C $\frac{2.sin\theta }{1-2co{s}^2\theta }$

Given: $\frac{1}{sin\theta +cos\theta }+\frac{1}{sin\theta -cos\theta }$

By taking LCM,

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

$=\frac{2sin\theta }{(si{n}^2\theta -co{s}^2\theta )}$

Using the identity, $si{n}^2\theta =1-co{s}^2\theta$

$=\frac{2sin\theta }{\left[\right(1-co{s}^2\theta )-co{s}^2\theta ]}$

$=\frac{2sin\theta }{(1-2co{s}^2\theta )}$