Question

integration of 1/cos(x-a)cos(x-b)

Answer 1

It is one of those peculiar integrals in the CBSE curriculum which would require you to memorise a crucial trick in order to quickly get through in a few steps.

1cos(x−a)cos(x−b)1cos⁡(x−a)cos⁡(x−b)

Main Step: Check that if you multiply and divide the integrand by sin(a−b)sin⁡(a−b), you would be able to simplify it with a bit of manipulation.

1sin(a−b)⋅sin((x−b)−(x−a))cos(x−a)cos(x−b)1sin⁡(a−b)⋅sin⁡((x−b)−(x−a))cos⁡(x−a)cos⁡(x−b)

Note that the term in the numerator can be expanded to obtain

sin(x−b)cos(x−a)−cos(x−b)sin(x−a)sin⁡(x−b)cos⁡(x−a)−cos⁡(x−b)sin⁡(x−a)

Then by cancelling like terms in the numerator and the denominator, we are only left with

1sin(a−b)⋅(tan(x−b)−tan(x−a))1sin⁡(a−b)⋅(tan⁡(x−b)−tan⁡(x−a))

The integration after which is basic.

1sin(a−b)⋅ln∣∣∣cos(x−a)cos(x−b)