Find all values of \(x\) such that \(0\leq x\leq 2\pi\) and \( \sin(x)-\cos(x) = \dfrac{1}{\sqrt{2}} .\)
+128460 sin (x) - cos(x) = 1/√2 square both sides
sin^2(x) - 2sin(x)cos(x) + cos^2x = 1/2
-2sin(x)cos(x) + 1 = 1/2
-2sin(x)cos(x) = -1/2
2sin(x)cos(x) = 1/2
sin (2x) = 1/2
sin (x) = 1/2 at pi/6 , 5pi/6 , 13pi/6 , 17pi/6
So
sin (2x) = 1/2 at pi/12 , 5pi/12 , 13pi/12 and 17pi/12
As the graph here shows, the only solutions occur at 5pi/12 and 13pi/12 : https://www.desmos.com/calculator/lknuil6pvy
[The other two solutions are extraneous because we squared the original equation ]
