SOLUTION: Prove that sin x+ cos x =sqrt(2)sin(x+pi/4) Please show the steps in detail, thank you

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Question 1086461: Prove that sin x+ cos x =sqrt(2)sin(x+pi/4)
Please show the steps in detail, thank you
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
The identity I'll use is

sin(x+y) = sin(x)*cos(y) + cos(x)*sin(y)

This identity will be used to transform the right side of the original equation like so

sin(x) + cos(x) = sqrt(2)*sin(x+pi/4)
sin(x) + cos(x) = sqrt(2) * [sin(x)*cos(pi/4)+cos(x)*sin(pi/4)]
sin(x) + cos(x) = sqrt(2) * [sin(x)*(sqrt(2)/2)+cos(x)*(sqrt(2)/2)]
sin(x) + cos(x) = sqrt(2)*(sqrt(2)/2) * [sin(x)+cos(x)]
sin(x) + cos(x) = ((sqrt(2)*sqrt(2))/2) * [sin(x)+cos(x)]
sin(x) + cos(x) = (sqrt(2*2)/2) * [sin(x)+cos(x)]
sin(x) + cos(x) = (sqrt(4)/2) * [sin(x)+cos(x)]
sin(x) + cos(x) = (2/2) * [sin(x)+cos(x)]
sin(x) + cos(x) = 1 * [sin(x)+cos(x)]
sin(x) + cos(x) = sin(x) + cos(x)

Note how the left side is kept the same. I only changed the right side. The last step confirms we have a true identity because we have the same expression on both sides.