Question

If cosx + sinx =√2cosx , prove that cosx-sinx =√2sinx

Answer 1

Answer:

Given,

cos x + sinx = √ 2 cos x , then (√2 -1 ) cos  x = sin x

on multiplying both sides by (√2+1) , we get

(√2+1)(√2-1) cos x  =  (√2+ 1) sin x

⇒ cos x = √2 sin x + sin x

⇒ cos x -sin x = √ 2 sin x

hence proved

Answer 2

Step-by-step explanation:

cosx + sinx = √2cosx

Squaring both sides

cos²x+sin²x+2sinxcosx=2cos²x

cos²x-sin²x=2sinxcosx → (1)

Now, (cosx - sinx)²=cos²x+sin²x-2sinxcosx

                             =cos²x + sin²x - cos²x + sin²x       [From (1)]

                             =2sin²x

So, cosx-sinx=√(2sin²x)

                     =±√2sinx