Question

IF cosx + sin X =✓2 cosx, prove that cosx - sinx =✓2sinx​

Answer 1

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Answer 2

Step-by-step explanation:

$\cos(x ) + \sin(x) = \sqrt{2} \cos(x ) - - - - - 1 \\ squaring \: both \: sides \: \\ { (\sin(x) + \cos(x)) }^{2} = 2 { \cos(x) }^{2} \\ { \sin(x) }^{2} + { \cos(x) }^{2} + 2 \sin(x) \cos(x) = 2 { \cos(x) }^{2} \\ 1 + 2 \sin(x) \cos(x) = 2 { \cos(x) }^{2} \\ 2 \sin(x) \cos(x) = 2 { \cos(x) }^{2} - 1 - - - - - 2\\ we \: know { \: \: \sin(x) }^{2} + { \cos(x) }^{2} = 1 \\ or { (\sin(x) - \cos(x) ) }^{2} + 2 \sin(x) \cos(x) = 1 \\ or {( \sin(x) - \cos(x)) }^{2} = 1 - 2 { \cos(x) }^{2} + 1 \\ or \cos(x ) - \sin(x) = \sqrt{2} \sqrt{(1 - { \cos(x) }^{2}) } \\ or \cos(x) - \sin(x) = \sqrt{2} \sin(x)$