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Trigonometric Identities — Maths STD 10 — Question

Gujarat BoardEnglish MediumSTD 10MathsTrigonometric Identities4 Marks
Question
Prove the following identities:
$\frac{\sin\theta+\cos\theta}{\sin\theta-\cos\theta}+\frac{\sin\theta-\cos\theta}{\sin\theta+\cos\theta}=\frac{2}{\big(1-2\cos^2\theta\big)}$
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Answer

$\text{LHS}=\frac{\sin\theta+\cos\theta}{\sin\theta-\cos\theta}+\frac{\sin\theta-\cos\theta}{\sin\theta+\cos\theta}$
$=\frac{(\sin\theta+\cos\theta)^2+(\sin\theta-\cos\theta)^2}{\sin^2\theta-\cos^2\theta}$
$=\frac{\sin^2\theta+\cos^2\theta+2\cos\theta\sin\theta+\sin^2\theta+\cos^2\theta+2\cos\theta\sin\theta}{1-\cos^2\theta-\cos^2\theta}$
$=\frac{1+1}{1-2\cos^2\theta}$
$=\frac{2}{\big(1-2\cos^2\theta\big)}$
$=\text{R.H.S.}$
$\therefore\text{R.H.S.}=\text{L.H.S.}$

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