Question

Integrate : $\frac{1}{\cos \left(x-a\right)\cos (x-b)}$

Answer 1

$\int \frac{1}{\cos (x-a)\cos (x-b)}dx$

$\int \frac{\sin (a-b)dx}{\sin (a-b)\cos (x-a)\cos (x-b)}$

$\int \frac{1}{\sin (a-b)}.\frac{\sin (x-b)+(a-x)}{\cos (x-a)\cos (x-b)}dx$

$\frac{1}{\sin (a-b)}\int \frac{\sin (a-b)\cos (a-x)+\cos (x-b)\sin (a-x)}{\cos (x-a)\cos (x-b)}dx$

$\frac{1}{\sin (a-b)}\int \tan (x-b)+\tan (x-a)dx$

$\frac{1}{\sin (a-b)}\left(\log \sec (x-b)-\log \sec (x-a)\right)+C$

$\frac{1}{\sin (a-b)}\log \left|\frac{\cos (x-a)}{\cos (x-b)}\right|+C$