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The Cosine Difference Formula is an essential tool in trigonometry. It allows us to express the cosine of the difference between two angles in terms of the sines and cosines of the individual angles. The formula is as follows: \[ \cos(a - b) = \cos a \cdot \cos b + \sin a \cdot \sin b \]This formula helps simplify expressions where angles are subtracted from each other, making complex trigonometric identities or calculations easier to solve. In practice, we use this formula by substituting the specific values of the angles into it. For example, to find \( \cos(x - \pi) \), we replace \( a \) with \( x \) and \( b \) with \( \pi \), resulting in the equation:\[ \cos(x - \pi) = \cos x \cdot \cos \pi + \sin x \cdot \sin \pi \] This substitution opens the path to evaluating and proving trigonometric identities like the one in our exercise.

Understanding the fundamental trigonometric values is crucial for applying formulas like the Cosine Difference Formula effectively. When working with the example \( \cos(x - \pi) \), knowing the values of \( \cos \pi \) and \( \sin \pi \) becomes important.

• \( \cos \pi \): This value is known to be \( -1 \). It is helpful to remember that \( \pi \) radians is equivalent to 180 degrees, where cosine values are negative and sine is zero.
• \( \sin \pi \): This value is 0, which is a standard outcome for sine at 180 degrees or \( \pi \) radians.

By substituting these values into our formula:\[ \cos(x - \pi) = \cos x \cdot (-1) + \sin x \cdot 0 = -\cos x \]This simplification underscores the importance of knowing key trigonometric values for angles like \( \pi \), \( \pi/2 \), etc.

Proving an identity involves showing that both sides of the equation are equivalent under the rules of algebra and trigonometry. In the given exercise, our goal was to show that \( \cos(x - \pi) = -\cos x \).Step-by-step, here's how the proof unfolds:

• Start with the formula: We began by applying the Cosine Difference Formula: \( \cos(x - \pi) = \cos x \cdot \cos \pi + \sin x \cdot \sin \pi \).
• Use known values: By substituting \( \cos \pi = -1 \) and \( \sin \pi = 0 \), the equation simplifies to \(-\cos x + 0 = -\cos x \).
• Conclude: This simplification adheres perfectly to the identity we're proving, confirming that our initial equation holds true.

Thus, we see that each step builds toward confirming the identity, validating the hypothesis using trig identities, known values, and arithmetic simplification. This methodical approach guarantees a solid understanding and defense of the given identity.