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Differentiation Using Substitution

Prove that: 1...

Question

Prove that: $\frac{1}{sin (x - a) sin (x - b)} = \frac{cot (x - a) - cot (x - b)}{sin (a - b)}$

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Solution

To Prove : $\frac{1}{sin (x - a) sin (x - b)} = \frac{cot (x - a) - cot (x - b)}{sin (a - b)}$

Taking R.H.S.

$\frac{cot (x - a) - cot (x - b)}{sin (a - b)}$

$\frac{\frac{cos (x - a)}{sin (x - a)} \frac{cos (x - b)}{sin (x - b)}}{sin (a - b)}$

$= \frac{sin (x - b) cos (x - a) - sin (x - a) cos (x - b)}{sin (x - a) sin (x - b) sin (a - b)}$

$= \frac{sin (x - b - x + a)}{sin (x - a) sin (x - b) sin (a - b)}$
$[∵ sin A cos B - sin B cos A]$

$= \frac{1}{sin (x - a) sin (x - b)}$

Hence proved.

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Differentiation Using Substitution

Standard XII Mathematics

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