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Trigonometric Identities — Maths STD 10 — Question

Maharashtra BoardEnglish MediumSTD 10MathsTrigonometric Identities2 Marks
Question
Prove the following trigonometric identities.
$\cot^2\text{A}\text{ cosec}^2\text{B}-\cot^2\text{B cosec}^2\text{A}=\cot^2\text{A}-\cot^2\text{B}$
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Answer

$\text{L.H.S}=\cot^2\text{A cosec}^2\text{B}-\cot^2\text{B cosec}^2\text{A}$
$=\cot^2\text{A}(1+\cot^2\text{B})-\cot^2\text{B}(1+\cot^2\text{B}) \big[\because \text{cosec}^2\theta=1+\cot^2\theta\big]$
$=\cot^2\text{A}+\cot^2\text{A}\cot^2\text{B}-\cot^2\text{B}-\cot^2\text{B}\cot^2\text{A}$
$=\cot^2\text{A}-\cot^2\text{B}=\text{R.H.S}$
$\therefore\ \text{L.H.S}=\text{R.H.S}$

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