Find all the solutions of the equation sinxcosx0 Please show

Find all the solutions of the equation sin(x)+cos(x)=0 Please show steps!!!

Solution

You could use trigonometric identities
sin x + cos x = 0
sin x + (1 - sin²(x)) = 0
sin x = -(1 - sin²(x))
sin²(x) = 1 - sin²(x)
2sin²(x) = 1
sin²(x) = ½
sin (x) = ±½
x = arcsin(±½) = /4, 3/4, 5/4, 7/4
Substitute in original equation to eliminate extraneous roots.
sin(/4) + cos(/4) = (½) + (½) 0
sin(3/4) + cos(3/4) = (½) + (-½) = 0
sin(5/4) + cos(5/4) = (-½) + (-½) 0
sin(7/4) + cos(7/4) = (-½) + (½) = 0
Therefore
x = 3/4, 7/4 (135°, 315°) are the roots
(in the interval 0 < x < 2)