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Trigonometric Ratios Of Multiple And Sub Multiple Angles — Maths STD 11 Science — Question

CBSE BoardEnglish MediumSTD 11 ScienceMathsTrigonometric Ratios Of Multiple And Sub Multiple Angles5 Marks
Question
Prove that:
$\frac{1}{\sin\text{(x}-\text{b})\sin\text{(x}-\text{b)}}=\frac{\cot\text{(x}-\text{a)}+\tan\text{(x}-\text{b)}}{\cos\text{(a}-\text{b)}}$
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Answer

$\text{L.H.S}=\frac{1}{\sin\text{(x}-\text{a)}\cos\text{(x}-\text{b)}}$
$=\frac{1}{\cos\text{(a}-\text{b)}}\Big[\frac{\sin\text{(a}-\text{b)}}{\sin\text{(x}-\text{b})\cos\text{(x}-\text{b)}}\Big]$
$=\frac{1}{\cos\text{(a}-\text{b)}}\Big[\frac{\cos\text{(x}-\text{b)-}(\text{x}-\text{b)}}{\sin\text{(x}-\text{a})\cos\text{(x}-\text{b)}}\Big]$
$=\frac{1}{\cos\text{(a}-\text{b)}}\Big[\frac{\cos\text{(x}-\text{b)cos}(\text{x}-\text{a)}+\sin\text{(x}-\text{b})\sin\text{(x}-\text{a)}}{\sin\text{(x}-\text{a})\sin\text{(x}-\text{b)}}\Big]$
$=\frac{1}{\cos\text{(a}-\text{b})}\Big[\frac{\cos\text{(x}-\text{b)}\cos\text{(x}-\text{a)}}{\sin\text{(x}-\text{b)}\cos\text{(x}-\text{b})}+\frac{\sin\text{(x}-\text{b)}\sin\text{(x}-\text{a)}}{\sin\text{(x}-\text{b)}\cos\text{(x}-\text{b})}\Big]$
$=\frac{\cot\text{(a}-\text{b)}+\tan\text{(x}-\text{b)}}{\cos\text{(a}-\text{b)}}$
$=\frac{1}{\sin\text{(a}-\text{b})}\Big[\cot\text{(x}-\text{b)}-\cot\text{(x}-\text{b)}\Big]$
$=\frac{\cot\text{(x}-\text{a)-}\cot\text{(x}-\text{b)}}{\sin\text{(a}-\text{b)}}$
$=\text{R.H.S}$
$\therefore\ \text{L.H.S}=\text{R.H.S}$
Hence proved.

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