Final Answer:
The expression "sin(4x) - cos(4x)" can be rewritten in terms of sine only as:
sin(4x + 45°)
Step-by-step explanation:
To express "sin(4x) - cos(4x)" solely in terms of sine, we can utilize trigonometric identities. First, let's write the expression in terms of sine and cosine of a single angle using the angle addition formula for sine:
sin(4x) - cos(4x) = sin(4x) - cos(90° - 4x)
Applying the cosine difference identity (cos(90° - θ) = sin(θ)), we get:
sin(4x) - sin(4x) = 2sin(4x)sin(45°)
Now, applying the double angle formula for sine (sin(2θ) = 2sin(θ)cos(θ)), we rewrite sin(4x) as sin(2 * 2x):
2sin(2x)cos(2x) = 2 * 2sin(x)cos(x)cos(2x)
Utilizing the identity for cosine double angle (cos(2θ) = 1 - 2sin²(θ)), we simplify further:
4sin(x)cos(x)(1 - 2sin²(x)) = 4sin(x)cos(x) - 8sin³(x)cos(x)
Finally, applying the identity for sin(45°) = √2 / 2 to sin(x) and cos(x), where x = 2x:
4 * (√2 / 2)(√2 / 2) - 8 * (√2 / 2)³ = 2 - 2 = 0
Thus, after simplification and substitution, the expression reduces to sin(4x + 45°). This identity allows the original expression to be rewritten solely in terms of sine.