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Prove the following trigonometric identities:$ \frac{1+\sin \theta}{\cos \theta}+\frac{\cos \theta}{1+\sin \theta}=2 \sec \theta $
To do:
We have to prove that \( \frac{1+\sin \theta}{\cos \theta}+\frac{\cos \theta}{1+\sin \theta}=2 \sec \theta \).
Solution:
We know that,
$\sec \theta=\frac{1}{\cos \theta}$.....(i)
$\cos ^{2} \theta+\sin^2 \theta=1$.......(ii)
Therefore,
$\frac{1+\sin \theta}{\cos \theta}+\frac{\cos \theta}{1+\sin \theta}=\frac{(1+\sin \theta)^2+(\cos^2 \theta)}{(\cos \theta)(1+\sin \theta)}$
$=\frac{1+2\sin \theta+\sin^2 \theta+\cos^2 \theta}{\cos \theta(1+\sin \theta)}$
$=\frac{2+2\sin \theta}{\cos \theta(1+\sin \theta)}$ [From (ii)]
$=\frac{2(1+\sin \theta)}{\cos \theta(1+\sin \theta)}$
$=\frac{2}{\cos \theta}$
$=2 \sec \theta$ [From (i)]
Hence proved.
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