Question

if cosx + sinx = √2 cosx then show cosx -sinx =√2 sinx

Answer 1

cos x+sinx= ✓2cosx
sinx=(✓2-1)cosx
multiply ✓2+1 both side
(✓2+1)sinx=(✓2+1)(✓2-1)cosx
✓2sinx+sinx=cos x
cosx-sinx=✓2sinx

Answer 2

Answer:

$if\: cosx + sinx = \sqrt{2} cosx\\ then \: cosx -sinx =\sqrt{2} sinx$

Step-by-step explanation:

$we \:have ,\\cosx+sinx=\sqrt{2}cosx---(1)$

On Squaring both sides of the equation,we get

$\implies (cosx+sinx)^{2}=\big(\sqrt{2}cosx\big)^{2}$

$\implies cos^{2}x+sin^{2}x+2cosxsinx=2cos^{2}x$

$\implies sin^{2}x=2cos^{2}x-cos^{2}x-2cosxsinx$

$\implies sin^{2}x=cos^{2}x-2cosxsinx$

Adding sin²x both sides of the equation,we get

$\implies 2sin^{2}x=cos^{2}x+sin^{2}x-2cosxsinx$

$\implies \big(\sqrt{2}sinx\big)^{2}x=(cosx-sinx)^{2}$

$\implies \sqrt{2}sinx=cosx-sinx$

Therefore,

$cosx-sinx=\sqrt{2}sinx$

•••♪