If cos θ + sin θ = √2 cos θ, find the value of (cos θ − sin θ).

cos θ + sin θ = √2 . cosθ       ----(1)

on squaring both side ,

cos² θ + sin² θ + 2cos θ.sin θ = 2 . cos²θ

2cos θ.sin θ =  cos²θ- sin²θ 

2cos θ.sin θ =  ( cosθ - sinθ ) . ( cosθ  + sinθ )

By using equation 1 ,

2cos θ.sin θ =  ( cosθ - sinθ ) . √2 . cosθ

cosθ - sinθ = \(\frac{2cos θ.sin θ}{√2 . cosθ}\)

cosθ - sinθ = √2 . sinθ