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Trigonometric Identities — Maths STD 10 — Question

CBSE BoardEnglish MediumSTD 10MathsTrigonometric Identities5 Marks
Question
Prove the following trigonometric identities.
$(\sec\text{A}-\text{cosec A})(1+\tan\text{A}+\cot\text{A})=\tan\text{A}\sec\text{A}-\cot\text{A}\text{ cosec A}$
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Answer

$\text{L.H.S}=(\sec\text{A}-\text{cosec A})(1+\tan\text{A}+\cot\text{A})$
$=\Big[\frac{1}{\cos\text{A}}-\frac{1}{\sin\text{A}}\Big]\Big[1+\frac{\sin\text{A}}{\cos\text{A}}+\frac{\cos\text{A}}{\sin\text{A}}\Big]$
$=\Big[\frac{\sin\text{A}-\cos\text{A}}{\sin\text{A}\cos\text{A}}\Big]\Big[\frac{\cos\text{A}\sin\text{A}+\sin^2\text{A}+\cos^2\text{A}}{\sin\text{A}\cos\text{A}}\Big]$
$=\frac{(\sin\text{A}-\cos\text{A})(\sin^2\text{A}+\cos\text{A}\sin\text{A}+\cos^2\text{A})}{\sin^2\text{A}\cos^2\text{A}}$
$=\frac{(\sin^3\text{A}-\cos^3\text{A})}{\sin^2\text{A}\cos^2\text{A}} \big[\therefore\ (\text{a}-\text{b})(\text{a}^2+\text{ab})+\text{b}=(\text{a}^3-\text{b}^2)\big]$
$=\text{R.H.S}=\tan\text{A}\sec\text{A}-\cot\text{A}\text{ cosec A}$
$=\frac{\sin\text{A}}{\cos\text{A}}\times\frac{1}{\cos\text{A}}-\frac{\cos\text{A}}{\sin\text{A}}\times\frac{1}{\sin\text{A}}$
$=\frac{\sin\text{A}}{\cos^2\text{A}}-\frac{\cos\text{A}}{\sin^2\text{A}}$
$=\frac{\sin^3\text{A}-\cos^3\text{A}}{\sin^2\text{A}\cos^2\text{A}}$
$\text{L.H.S}=\text{R.H.S}$
Hence proved.

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