Question

If sin x + COS X = √2 cosx
then cos X-sin x=​

Answer 1

sin x+cos x= √2 cos x

cos x- sin x= √2 cos x

squaring on both sides, we get

cos2x + sin2x+2 cosx sinx= 2cos2 x

=2cosx sinx= cos2x - sin2x

2 sinx cosx= (cosx+sinx)(cosx-sinx)

=2 sinx cos x/(cosx- sinx) = cos x+ sinx

2sinx cos x /(cos x- sinx) = √2 cos x

2√2 sinx/ cosx- sinx= √2*√2

2√2 sinx/ cosx-sinx= 2

therefore cos x - sin x= √2 sinx

thank you