Explanation

(c)
$\int_{}^{}\frac{\sec{x\ cosec\ x}}{2\cot{x - \sec{x\ cosec\ x}}}\ dx$
$= \int_{}^{}\frac{\frac{1}{\cos{x\sin x}}}{\frac{2\cos x}{\sin x} - \frac{1}{\sin{x\cos x}}}\ dx$
$= \int_{}^{}\frac{dx}{2\cos^{2}{x - 1}}$
$= \int_{}^{}\frac{dx}{\cos{2x}} = \int_{}^{}{\sec{2x}}\ dx$
$= \frac{1}{2}\log{|\sec{2x + \tan{2x| + c}}}$