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Trigonometry — Mathematics STD 10 — Question

ICSE BoardEnglish MediumSTD 10MathematicsTrigonometry4 Marks
Question
If $\sec \theta + \tan \theta = p,$ show that $\frac{p^2-1}{p^2+1}=\sin \theta$
✓

Answer

We have,
$=\frac{\sec ^2 \theta+\tan ^2 \theta+2 \sec \theta \tan \theta-1}{\sec ^2 \theta+\tan ^2 \theta+2 \sec \theta \tan \theta+1}$
$ =\frac{\left(\sec ^2 \theta-1\right)+\tan ^2 \theta+2 \sec \theta \tan \theta}{\sec ^2 \theta+2 \sec \theta \tan \theta+\left(1+\tan ^2 \theta\right)} $
$ =\frac{\tan ^2 \theta+\tan ^2 \theta+2 \sec \theta \tan \theta}{\sec ^2 \theta+2 \sec \theta \tan \theta+\sec ^2 \theta} $
$ =\frac{2 \tan ^2 \theta+2 \tan \theta \sec \theta}{2 \sec ^2 \theta+2 \sec \theta \tan \theta} $
$ =\frac{2 \tan \theta(\tan \theta+\sec \theta)}{2 \sec \theta(\sec \theta+\tan \theta)}$
$=\frac{\tan \theta}{\sec \theta}=\frac{\sin \theta}{\cos \theta \sec \theta} $
$=\sin \theta=\text { RHS }$

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