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Trigonometric Identities — Maths STD 10 — Question

CBSE BoardEnglish MediumSTD 10MathsTrigonometric Identities3 Marks
Question
Prove the following trigonometric identities.
$\frac{\tan\text{A}}{(1+\tan^2\text{A})^2}+\frac{\cot\text{A}}{(1+\cot^2\text{A})}=\sin\text{A}\cos\text{A}$
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Answer

We have to prove $\frac{\tan\text{A}}{(1+\tan^2\text{A})^2}+\frac{\cot\text{A}}{(1+\cot^2\text{A})}=\sin\text{A}\cos\text{A}$
We know that, $\sin^2\text{A}+\cos^2\text{A}=1$
So,
$\text{L.H.S}=\frac{\tan\text{A}}{(1+\tan^2\text{A})^2}+\frac{\cot\text{A}}{(1+\cot^2\text{A})}$
$=\frac{\tan\text{A}}{(\sec^2\text{A})^2}+\frac{\cot\text{A}}{(\text{cosec}^2\text{A})^2}$
$=\frac{\tan\text{A}}{\sec^4\text{A}}+\frac{\cot\text{A}}{\text{cosec}^4\text{A}}$
$=\frac{\frac{\sin\text{A}}{\cos\text{A}}}{\frac{1}{\cos^4\text{A}}}+\frac{\frac{\cos\text{A}}{\sin\text{A}}}{\frac{1}{\sin^4\text{A}}}$
$=\frac{\sin\text{A}\cos^4\text{A}}{\cos\text{A}}+\frac{\cos\text{A}\sin^4\text{A}}{\sin\text{A}}$
$=\sin\text{A}\cos^3\text{A}+\cos\text{A}\sin^3\text{A}$
$=\sin\text{A}\cos\text{A}(\cos^2\text{A}+\sin^2\text{A})$
$=\sin\text{A}\cos\text{A}=\text{R.H.S}$
$\therefore\ \text{L.H.S}=\text{R.H.S}$

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