Math, asked by kamal162, 1 year ago

if sec theta +tan theta =p. prove that sin theta = (p2-1) / (p2 + 1)

Answers

Answered by Sakshi15403
562
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Answered by mysticd
246

Solution:

We have ,

sec\theta + tan\theta = p ----(1)

Now,

sec^{2}\theta-tan^{2}\theta = 1

\implies (sec\theta + tan\theta)(sec\theta - tan\theta)=1

\implies p\times (sec\theta - tan\theta)=1

\implies sec\theta + tan\theta = \frac{1}{p}--(2)

Adding and subtracting (1) and (2) , we get

 2sec\theta = \frac{(p^{2}+1)}{p} ---(3)

 2tan\theta = \frac{(p^{2}-1)}{p} ---(4)

on dividing equation (4) by (3), we get

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

\implies \frac{\frac{sin\theta}{cos\theta}}{\frac{1}{cos\theta}}= \frac{\frac{(p^{2}-1)}{p}}{\frac{(p^{2}+1)}{p}}

After cancellation, we get

sin\theta = \frac{\frac{(p^{2}-1)}{p}}{\frac{(p^{2}+1)}{p}}

Hence, proved .

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