Final answer:
To write sin(2x) - cos(2x) in terms of sine only, apply the trigonometric identity cos(θ) = sin(π/2 - θ). Then use the trigonometric identity sin(a) - sin(b) = 2cos((a+b)/2)sin((a-b)/2) to simplify the expression.
Step-by-step explanation:
To write the expression sin(2x) - cos(2x) in terms of sine only, we can use the trigonometric identity cos(θ) = sin(π/2 - θ). Applying this identity to the expression, we get:
sin(2x) - cos(2x) = sin(2x) - sin(π/2 - 2x)
Using the trigonometric identity sin(a) - sin(b) = 2cos((a+b)/2)sin((a-b)/2), we can simplify the expression further:
sin(2x) - sin(π/2 - 2x) = 2cos((2x+(π/2 - 2x))/2)sin((2x-(π/2 - 2x))/2)
Simplifying the angles in the cosine and sine functions, we get:
sin(2x) - cos(2x) = 2cos(π/4)sin(π/4)
Finally, simplifying the trigonometric values, we have:
sin(2x) - cos(2x) = 2(sqrt(2)/2)(sqrt(2)/2) = 2/2 = 1