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Trigonometric Identities — Maths STD 10 — Question

CBSE BoardEnglish MediumSTD 10MathsTrigonometric Identities5 Marks
Question
Prove the following identities:
$\frac{\cos\theta\text{ cosec }\theta-\sin\theta\sec\theta}{\cos\theta+\sin\theta}=\text{cosec }\theta-\sec\theta$
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Answer

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

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