Question

Find the general solution of : cos x - sin x = 1.

Answer 1

cos x - sin x = 1

Dividing by `sqrt (1^2 + (-1)) = sqrt2`

`1/sqrt 2  "cos x" - 1/ sqrt 2  "sin x" = 1/ sqrt2`

`"cos"  pi/4 "cos  x" - "sin" pi/4 "sin  x" = 1/ sqrt2`

`"cos" ("x" + pi/4) = "cos" pi/4`   ...(i)

The general solution of cos θ = cos α is θ = 2nπ ± `pi/4 ; n ∈ z`

∴ The genera; solution of equation (i) given by 

`"x" + pi/4 = 2"n"pi +- pi/4 ; n ∈ z`

`"x" = 2 n pi ; x = 2 n pi - pi/2 ; "n" ∈ z`