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If secθ + tanθ = p, show that (p^2−1)/(p^2+1)=sinθ - CBSE Class 10 - Mathematics

Question

If secθ + tanθ = p, show that (p^{2}-1)/(p^{2}+1)=\sin \theta

Solution

We have,

=(\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{(\sec ^{2}\theta -1)+\tan ^{2}\theta +2\sec \theta \tan\theta }{\sec ^{2}\theta +2\sec \theta \tan \theta +(1+\tan^{2}\theta )

=(\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 }

= sinθ = RHS

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Solution If secθ + tanθ = p, show that (p^2−1)/(p^2+1)=sinθ Concept: Trigonometric Identities.
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