Question

Is LHS=RHS?

$\frac{1}{cosec\theta +cot\theta }$ - $\frac{1}{\sin \theta }$ = $\frac{1}{\sin \theta }$ - $\frac{1}{cosec\theta -cot\theta }$

Say true or false.

• A. True
• B. False
• C. Ambiguous
• D. Data insufficient

Answer 1

The correct option is A True

$\frac{1}{cosec\theta +cot\theta }-\frac{1}{sin\theta }$
$=\frac{sin\theta }{1+cos\theta }-\frac{1}{sin\theta }$
$=\frac{si{n}^2\theta -1-cos\theta }{sin\theta (1+cos\theta )}$
$=\frac{-co{s}^2\theta -cos\theta }{sin\theta (1+cos\theta )}$
$=\frac{-cos\theta }{sin\theta }$
$=-cot\theta$
$=LHS$
$RHS$
$=\frac{1}{sin\theta }-\frac{1}{cosec\theta -cot\theta }$
$=\frac{1}{sin\theta }-\frac{sin\theta }{1-cos\theta }$
$=\frac{1-cos\theta -si{n}^2\theta }{sin\theta (1-cos\theta )}$
$=\frac{co{s}^2\theta -cos\theta }{sin\theta (1-cos\theta )}$
$=\frac{-cos\theta }{sin\theta }$
$=-cot\theta$
Hence
$LHS=RHS$
Hence proved.