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Trigonometric Identities — Maths STD 10 — Question

CBSE BoardEnglish MediumSTD 10MathsTrigonometric Identities2 Marks
Question
Prove the following trigonometric identities.
$\frac{\text{cosec A}}{\text{cosec A}-1}+\frac{\text{cosec A}}{\text{cosec A}+1}=2\sec^2\text{A}$
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Answer

$\text{L.H.S}=\frac{\text{cosec A}}{\text{cosec A}-1}+\frac{\text{cosec A}}{\text{cosec A}+1}$
$=\frac{\text{cosec A}(\text{cosec A+1})+\text{cosec A}(\text{cosec A}+1)}{(\text{cosec A}-1)(\text{cosec}+1)}$
$=\frac{\text{cosec}^2\text{A}+\text{cosec A}+\text{cosec}^2\text{A}-\text{cosec A}}{\text{cosec}^2-1}$
$=\frac{2\text{cosec}^2\text{A}}{\cot^2\text{A}}$
$=\frac{2\times\sin^2\text{A}}{\sin^2\text{A}\cos^2\text{A}}$
$=2\sec^2\text{A}$
$=\text{R.H.S}$
$\therefore\ \text{L.H.S}=\text{R.H.S}$

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