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Trigonometric Identities — Maths STD 10 — Question

Maharashtra BoardEnglish MediumSTD 10MathsTrigonometric Identities2 Marks
Question
Prove the following trigonometric identities.
$(\sec^2\theta-1)(\text{cosec}^2\theta-1)=1$
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Answer

$\text{L.H.S}=(\sec^2\theta-1)(\text{cosec}^2\theta-1)$
$ [\because\ 1+\tan^2\theta=\sec^2\theta,1+\cot^2\theta=\text{cosec}^2\theta]$
$=(1+\tan^2\theta-1)(1+\cot^2\theta-1)$
$=\tan^2\theta\times\cot^2\theta$
$=\tan^2\theta\times\frac{1}{\tan^2\theta}$
$=1=\text{R.H.S}$
$\therefore\ \text{L.H.S}=\text{R.H.S}$

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