int_{pi /4}^{3pi /4} frac{dx}{1 + cos x} is e | vedclass

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(A) Let $I = \int_{\pi /4}^{3\pi /4} \frac{dx}{1 + \cos x}$.Using the identity $1 + \cos x = 2\cos^2(x/2)$,we have:$I = \int_{\pi /4}^{3\pi /4} \frac{

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$2$

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(A) Let $I = \int_{\pi /4}^{3\pi /4} \frac{dx}{1 + \cos x}$.Using the identity $1 + \cos x = 2\cos^2(x/2)$,we have:$I = \int_{\pi /4}^{3\pi /4} \frac{dx}{2\cos^2(x/2)} = \frac{1}{2} \int_{\pi /4}^{3\pi /4} \sec^2(x/2) dx$.Integrating $\sec^2(x/2)$,we get:$I = \frac{1}{2} [2 	an(x/2)]_{\pi /4}^{3\pi /4} = [	an(x/2)]_{\pi /4}^{3\pi /4}$.Evaluating at the limits:$I = 	an(3\pi /8) - 	an(\pi /8)$.Using $	an(3\pi /8) = \cot(\pi /8)$:$I = \cot(\pi /8) - 	an(\pi /8) = \frac{\cos(\pi /8)}{\sin(\pi /8)} - \frac{\sin(\pi /8)}{\cos(\pi /8)} = \frac{\cos^2(\pi /8) - \sin^2(\pi /8)}{\sin(\pi /8)\cos(\pi /8)}$.Using $\cos(2	heta) = \cos^2	heta - \sin^2	heta$ and $2\sin	heta\cos	heta = \sin(2	heta)$:$I = \frac{\cos(\pi /4)}{\frac{1}{2}\sin(\pi /4)} = 2 \cot(\pi /4) = 2(1) = 2$.

$\int_{\pi /4}^{3\pi /4} \frac{dx}{1 + \cos x}$ is equal to

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