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Question
Write the function in the simplest form: `tan^(-1) ((cos x - sin x)/(cos x + sin x)) `,` 0 < x < pi`
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Solution
`tan^(-1) (cos x - sin x)/(cos x + sin x)`
Dividing cos x inside
`= tan^(-1) [((cos x- sin x)/cos x)/((cos x + sinx)/cosx)]`
`= tan^(-1) [(cos x/cos x - sin x/cos x)/(cos x/cos x + sin x/cosx)]`
`= tan^(-1) (1 - tan x)/(1 + tan x)`
`= tan^(-1) [(1 - tan x)/(1+1. tan x)]`
`= tan^(-1) [(tan pi/4 - tan x)/(1 + tan pi/4 . tan x)]` (As tan `pi/4` = 1)
`= tan^(-1) tan (pi/4 - x)`
`= pi/4 - x`
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