Cos(x-7pi)-1

To simplify the expression \( \cos(x - 7\pi) - 1 \), we can use the cosine function's periodicity and properties.

1. The cosine function has a period of \( 2\pi \). This means that \( \cos(\theta) = \cos(\theta + 2k\pi) \) for any integer \( k \).

2. We can rewrite \( 7\pi \) in terms of a multiple of \( 2\pi \): \[ 7\pi = 3 \cdot 2\pi + \pi \] Thus, we have: \[ \cos(x - 7\pi) = \cos(x - (6\pi + \pi)) = \cos(x - \pi) \]

3. The cosine function has the property that \( \cos(x - \pi) = -\cos(x) \).

4. Substitute back into the original expression: \[ \cos(x - 7\pi) - 1 = \cos(x - \pi) - 1 \] \[ = -\cos(x) - 1 \]

Therefore, the expression simplifies to: \[ -\cos(x) - 1 \] Or you could write it as: \[ -1 - \cos(x) \]