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Question
Integrate the function:
`1/(cos (x+a) cos(x+b))`
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Solution
Let I = `1/(cos (x + a) cos (x + b)) dx`
`= 1/(sin (a - b)) int (sin [(x + a) - (x + b)])/(cos (x + a) cos (x + b))`dx
`= 1/(sin (a + b)) int (sin (x + a) cos (x + b) - cos (x + a) sin (x + b))/(cos (x + a) cos (x + b)) dx`
`= 1/(sin (a - b)) int [(sin (x + a))/(cos (x + a)) - (sin (x + b))/(cos (x + b))] dx`
`= 1/(sin (a - b)) int [tan (x + a) - tan (x + b)] dx`
`= 1/(sin (a - b)) = [log |cos (x + a)| - log |cos (x + b)|] + C`
`= 1/(sin (a - b)) = [log |sec(x + a)| - log |sec(x + b)|] + C`
`= 1/(sin (a - b)) log |(sec(x + a))/(sec(x + b))| + C`
or `= 1/(sin (a - b)) log |(cos (x + a))/(cos (x + b))| + C`
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