Rewrite the equation:
f(x) = (sin 2x - cos 2x) - (sin x - cos x) = 0
Use trig identity:
#sin a - cos a = sqrt2.sin (a - pi/4)#
we get:
#sin 2x - cos 2x = sqrt2.sin (2x - pi/4)#, and
#sin x - cos x = - sqrt2.sin (x - pi/4)#
#f(x) = sqrt2.sin (2x - pi/4) - sqrt2.sin (x - pi/4)#, -->
#sin (2x - pi/4) + sin (x - pi/4) = 0#
Apply the trig identity:
#sin a + sin b = 2sin ((a + b)/2).cos ((a - b)/2)#
#sin (2x - pi/4) + sin (x - pi/4) = sin ((3x)/2 - pi/2).cos x = 0#
There are 2solutions:
a. cos x = 0 --> #x = pi/2 and x = (3pi)/2#
b. #sin ((3x)/2 - pi/2) = 0#
#(3x)/2 = pi# --> #3x = 2pi #--> #x = (2pi)/3#
#(3x)/2 = 2pi# --># 3x = 4pi #--> #x = (4pi)/3#
Answers for #(0, 2pi)#:
#pi/2; (3pi)/2; (2pi)/3; and (4pi)/3#
