Question

Cosx+sinx=√2cos x show that cos x-sinx=√2 sinx

Answer 1

Cosx + since=√2cosx
(Cosx+sinx)^2=2cosx
Cos^2x-sim^2x=2cosxsinx
Cosx-sinx=✓2sinx

Answer 2

Hi,

Cosx + sinx = √2cosx-----( 1 )

Square both sides of equation ( 1 ),

( Cosx + sinx )² = ( √2cosx )²

Cos ² x + sin² x + 2sinxcosx = 2cos²x

1 + 2sinxcosx = 2cos²x

2sinx cosx = 2cos²x -1 ----( 2 )

(Cosx-sinx)² = (cosx + sinx)²-4sinxcosx

= ( √2cosx)² - 2(2cos²x-1)

= 4cos²x - 4cos²x + 2

= 2

Therefore ,

Cosx - sinx = √2

I hope this helps you.

:)