sin(a + b) = sin(a)cos(b) + cos(a)sin(b)
sin(x + 45) = sin(x)cos(45) + cos(x)sin(45) = sin(x)*sq rt(2)/2 + cos(x)sq rt(2)/2 =
= [sq rt(2)/2]*[sin(x) + cos(x)]
This equals sq rt(2)cos(x); therefore:
[sq rt(2)/2]*[sin(x) + cos(x)] = sq rt(2)cos(x)
sq rt(2) is multiplying both sides of the equation so it gets canceled out:
[sin(x) + cos(x)] / 2 = cos(x)
sin(x) + cos(x) = 2cos(x)
sin(x) = cos(x)
This is true for x = 45 degrees and x = 225 degrees (examining chart with the four quadrants).
Finally, we see if both values of x satisfy the original equation.
For x = 45 degrees:
sin (x + 45 degrees) = (square root 2)cosx
sin(45 + 45) = sq rt(2)cos(45) ----->
sin(90) = sq rt(2)*[sq rt(2) / 2]
1 = 2/2 -----> 1 = 1 (good)
Now for x = 225
sin (x + 45 degrees) = (square root 2)cosx
sin(225 + 45) = sq rt(2)cos(225)
sin(270) = sq rt(2)*[-sq rt(2)/2]
-1 = -2/2 -----> -1 = -1 (good)
The values for x are 45 degrees and 225 degrees. I hope this helps...