Verify the Identity cos(x-pi)=-cos(x)

Problem

cos(x−π)=−cos(x)

Solution

1. Identify the appropriate trigonometric identity to use, which is the cosine difference formula: cos(α−β)=cos(α)*cos(β)+sin(α)*sin(β)

2. Substitute the values α=x and β=π into the formula.

3. Expand the left side of the equation using the identity.

cos(x−π)=cos(x)*cos(π)+sin(x)*sin(π)

4. Evaluate the trigonometric constants cos(π) and sin(π)

cos(π)=−1

sin(π)=0

5. Simplify the expression by substituting these constants back into the expanded equation.

cos(x−π)=cos(x)*(−1)+sin(x)*(0)

cos(x−π)=−cos(x)+0

cos(x−π)=−cos(x)

Final Answer

cos(x−π)=−cos(x)

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