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Trigonometric Identities — Maths STD 10 — Question

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

$\text{L.H.S.}=\frac{\tan\theta}{\big(1+\tan^2\theta\big)}+\frac{\cot\theta}{\big(1+\cot^2\theta\big)}$
$=\frac{\tan\theta}{\big(\sec^2\theta\big)^2}+\frac{\cot\theta}{\big(\text{cosec}^2\theta\big)^2}$
$=\frac{\sin\theta}{\cos\theta}\times\frac{1}{\sec^4\theta}+\frac{\cos\theta}{\sin\theta}\times\frac{1}{\text{cosec}^4\theta}$
$=\frac{\sin\theta}{\cos\theta}\times\cos^4\theta+\frac{\cos\theta}{\sin\theta}\times\sin^4\theta$
$=\sin\theta\cos^3\theta+\cos\theta\sin^3\theta$
$=\sin\theta\cos\theta\big(\cos^2\theta+\sin^2\theta\big)$
$=\sin\theta\cos\theta$
$=\text{R.H.S.}$
$\therefore\text{R.H.S.}=\text{L.H.S.}$

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