Show, using complex numbers, that sin(x)+cos(x)=(√2)cos(x-∏/4)
The Attempt at a Solution
I was given the hint that sin(x)=Re(-ie^(ix)), but have thus far not been able to determine its usefullness. I have tried squaring the expression, which (after simplification) yields:
but cannot seem to go further. I assume that the crux of the solution lies in fully expressing sin(x)+cos(x) as purely cos in terms of complex exponentials, but everything I try just brings me back to the original expression. Am I missing a fundamental equivalence between either of these trigonometric functions and a complex number? Any help would be greatly appreciated, thank-you in advance.
(Obviously all work done in trying to solve this problem involves the assumption that I do not know the final answer)