Question

Solve by doubling both side:
cos x - sin x = sqrt 2/2

Answer 1

that's (sqrt 2)/2

Answer 2

2(cos x - sin x) = sqrt(2)
( 2(cos x - sin x) )^2 = 2
4(cos^2 x - 2sinxcosx + sin^2) = 2
sin^2x + cos^2x - 2sinxcosx = 1/2
1 - 2sinxcosx = 1/2
-sinxcosx = -1/2
sinxcosx = 1/2

Answer 3

2sinxcosx = 1
sin2x = 1

Answer 4

@mathstudent55

Answer 5

would 2sinxcos and 2cosxsinx the same thing?

Answer 6

just switched around a bit

Answer 7

Yes.
2sinxcosx = sin2x. This is one of the double angle identities.
Now that you have sin2x = 1, that will give you 2x easily.
Then divide the result by 2 to get x.
Remember that if you are solving this in an interval, such as [0, 2pi], you have one answer. If you are solving it in general, then you have to add a term with n (any integer), to account for the periodic repetition of the solution.