Question

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

Answer 1

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

Answer 2

Answer:

Hence proved that $\cos x - \sin x = \pm \sqrt { 2 } \sin x$

To prove:

$\cos x - \sin x = \pm \sqrt { 2 } \sin x$

Solution:

By the given identity, we know that

$\cos x + \sin x = \sqrt { 2 } \cos x$

$\begin{array} { c } { \sin x = \sqrt { 2 } \cos x - \cos x } \\\\ { \sin x = ( \sqrt { 2 } - 1 ) \cos x } \\\\ { ( \sqrt { 2 } + 1 ) ( \sqrt { 2 } - 1 ) \cos x = ( \sqrt { 2 } + 1 ) \sin x } \end{array}$

We know that,

$( a + b ) ( a - b ) = a ^ { 2 } - b ^ { 2 }$

$\begin{array} { c } { \left( ( \sqrt { 2 } ) ^ { 2 } - ( 1 ) ^ { 2 } \right) \cos x = ( \sqrt { 2 } + 1 ) \sin x } \\\\ { ( 2 - 1 ) \cos x = ( \sqrt { 2 } + 1 ) \sin x } \\\\ { \cos x = \sqrt { 2 } \sin x + \sin x } \\\\ { \cos x - \sin x = \sqrt { 2 } \sin x } \end{array}$

Hence Proved that $\cos x - \sin x = \pm \sqrt { 2 } \sin x$