Question

Solve for x: sinx-cosx=√2

Answer 1

sinx - cosx =sqrt(2)
Taking square on both sides:
(sinx - cosx)^2 =sqrt(2)^2
sin^2(x) -2cos(x)sin(x) + cos^2(x) = 2
Rearranging the equation:
sin^2(x)+cos^2(x) -2cos(x)sin(x)=2
As,
sin^2(x)+cos^2(x) = 1
So,
1-2sinxcosx=2
 1-1-2sinxcosx=2-1
 -2sinxcosx = 1
Using Trignometric identities:
-2(0.5(sin(x+x)+sin(x-x))=1 
-sin2x+sin0=1
As,
sin 0 = 0
So,
sin2x+0 = -1
sin2x = -1
2x=-90 degrees + t360
Dividing by 2 on both sides:
x=-45 degrees + t180
or 2x=270 degrees +t360
x= 135 degrees + t180 where t is integer