Question

$\int \frac{{sec}^{2}x}{{cosec}^{2}x}dx$

Answer 1

$\int \frac{{\sec }^{2}x}{{cosec}^{2}x}dx$

$=\int \frac{1+{\tan }^{2}x}{1+{\cot }^{2}x}dx$

$=\int \frac{1+{\tan }^{2}x}{1+1/{\tan }^{2}x}dx$

$=\int {\tan }^{2}x\frac{(1+{\tan }^{2}x)}{(1+{\tan }^{2}x)}dx$

$=\int {\tan }^{2}xdx$

$=\int ({\sec }^{2}x-1)dx$

$=\int {\sec }^{2}xdx-\int dx$

$=tanx-x+c$