Explanation

(b)
Let $I = \int_{0}^{\frac{\pi}{2}}\frac{\cos{3x + 1}}{2\cos{x - 1}}\ dx$
$= \int_{0}^{\frac{\pi}{2}}\frac{\cos{3x - \cos\frac{3\pi}{3}}}{2\left( \cos{x - \cos{\frac{\pi}{3}\ }} \right)}\ dx$
$= \int_{0}^{\frac{\pi}{2}}\frac{\left( 4\cos^{3}x - 3\cos{x) - (4\cos^{3}\frac{\pi}{3}} - 3\cos\frac{\pi}{3} \right)}{2\left( \cos x - \cos\frac{\pi}{3} \right)}dx$
$= 2\int_{0}^{\frac{\pi}{2}}{\left( \frac{\cos^{3}x - \cos^{3}\frac{\pi}{3}}{\cos x - \cos\frac{\pi}{3}} \right)dx - \frac{3}{2}\int_{0}^{\frac{\pi}{2}}\left( \frac{\cos{x - \cos\frac{\pi}{3}}}{\cos x - \cos\frac{\pi}{3}} \right)}dx$
$= 2\int_{0}^{\frac{\pi}{2}}\left( \cos^{2}x + \cos^{2}\frac{\pi}{3} + \cos x\cos\frac{\pi}{3} \right)dx - \frac{3}{2}\int_{0}^{\frac{\pi}{2}}1dx$
$= \int_{0}^{\frac{\pi}{2}}\left( 1 + \cos{2x} + \frac{1}{2} + \cos x \right)dx - \frac{3\pi}{4}$
$= \frac{3\pi}{4} + 1 - \frac{3\pi}{4} = 1$