Question

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

Answer 1

Hope it will help you..!..:+)

Answer 2

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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