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Trigonometric Functions — Maths STD 11 Science — Question

CBSE BoardEnglish MediumSTD 11 ScienceMathsTrigonometric Functions5 Marks
Question
Prove the following identities:
$\frac{\tan^3\text{x}}{1+\tan^2\text{x}}+\frac{\cot^3\text{x}}{1+\cot^2\text{x}}=\frac{1-2\sin^2\text{x}\cos^2\text{x}}{\sin\text{x}\cos\text{x}}$
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Answer

$\text{L.H.S}=\frac{\tan^{3}\text{x}}{1+\tan^{2}\text{x}}+\frac{\cot^{3}\text{x}}{1+\cot^{2}\text{x}}$
$=\frac{\sin^{3}\text{x}}{\cos^{3}\text{x}\bigg(1+\frac{\sin^{2}\text{x}}{\cos^{2}\text{x}}\bigg)}+\frac{\cos^{3}\text{x}}{\sin^{3}\text{x}\bigg(1+\frac{\cos^{2}\text{x}}{\sin^{2}\text{x}}\bigg)}$$$ $\Big(\because\tan\text{x}=\frac{\sin\text{x}}{\cos\text{x}}\text{ and }\cot\text{x}=\frac{\cos\text{x}}{\sin\text{x}}\Big)$
$=\frac{\sin^{3}\text{x}\cos^{2}\text{x}}{\cos^{3}\text{x}\big(\cos^{2}\text{x}+\sin^{2}\text{x}\big)}+\frac{\cos^{3}\text{x}\sin^{2}\text{x}}{\sin^{3}\text{x}\big(\sin^{2}\text{x}+\cos^{2}\text{x}\big)}$$$
$=\frac{\sin{3}\text{ x}}{\cos\text{x}}+\frac{\cos^{3}\theta}{\sin\text{x}}$ $\big(\because\cos^{2}\text{x}+\sin^{2}\text{x}=1\big)$
$=\frac{\sin^{4}\text{x}+\cos^{4}\text{x}}{\sin\text{x}\cos\text{x}}$
$=\frac{\big(\sin^{2}\text{x}\big)^{2}+\big(\cos^{2}\text{x}\big)^{2}+2\sin^{2}\text{x}\cos^{2}\text{x}-2\sin^{2}\text{x}\cos^{2}\text{x}}{\sin\text{x}\cos\text{x}}$ $\big(\text{adding and subtracting }2\sin^{2}\text{x}\cos^{2}\text{x}\big)$
$=\frac{\big(\sin^{2}\text{x}+\cos^{2}\text{x}\big)^{2}-2\sin^{2}\text{x}\cos^{2}\text{x}}{\sin\text{x}\cos\text{x}}$
$=\frac{1^{2}-2\sin^{2}\text{x}\cos^{2}\text{x}}{\sin\text{x}\cos\text{x}}$ $\big(\because\sin^{2}\text{x}+\cos^{2}\text{x}=1\big)$
$=\frac{1-2\sin^{2}\text{x}\cos^{2}\text{x}}{\sin\text{x}\cos\text{x}}$
$=\text{R.H.S}$
$\text{Proved}$

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