Introduction:

In this problem, we are asked to find the value of sin(45°θ) - cos(45°θ), where θ represents an angle. We will break down the problem step by step to calculate the value.

Calculating sin(45°θ):
To calculate sin(45°θ), we need to find the sine of an angle that equals 45° multiplied by θ. Let's break it down further:

1. Convert the angle to radians:
- 45° * (π/180) = π/4 * θ

2. Use the trigonometric identity for sine:
- sin(a + b) = sin(a)cos(b) + cos(a)sin(b)

3. Apply the identity to our equation:
- sin(π/4 * θ) = sin(π/4)cos(θ) + cos(π/4)sin(θ)

4. Substitute the values of sin(π/4) and cos(π/4):
- sin(π/4) = cos(π/4) = √2/2

5. Simplify the equation:
- sin(π/4 * θ) = (√2/2)cos(θ) + (√2/2)sin(θ)
- sin(π/4 * θ) = (√2/2)(cos(θ) + sin(θ))

Calculating cos(45°θ):
Now, let's calculate cos(45°θ) using a similar approach:

1. Convert the angle to radians:
- 45° * (π/180) = π/4 * θ

2. Use the trigonometric identity for cosine:
- cos(a + b) = cos(a)cos(b) - sin(a)sin(b)

3. Apply the identity to our equation:
- cos(π/4 * θ) = cos(π/4)cos(θ) - sin(π/4)sin(θ)

4. Substitute the values of cos(π/4) and sin(π/4):
- cos(π/4) = sin(π/4) = √2/2

5. Simplify the equation:
- cos(π/4 * θ) = (√2/2)cos(θ) - (√2/2)sin(θ)
- cos(π/4 * θ) = (√2/2)(cos(θ) - sin(θ))

Final Calculation:
Now that we have both sin(45°θ) and cos(45°θ), let's find their difference:

sin(45°θ) - cos(45°θ) = (√2/2)(cos(θ) + sin(θ)) - (√2/2)(cos(θ) - sin(θ))

Expanding the equation, we get:

sin(45°θ) - cos(45°θ) = (√2/2)cos(θ) + (√2/2)sin(θ) - (√2/2)cos(θ) + (√2/2)sin(θ)

The (√2/2)cos(θ) and -

√2 Sin theta