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

Problem

cos(π−x)=−cos(x)

Solution

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

2. Substitute the values A=π and B=x into the difference formula.

cos(π−x)=cos(π)*cos(x)+sin(π)*sin(x)

3. Evaluate the trigonometric constants using the unit circle, where cos(π)=−1 and sin(π)=0

cos(π−x)=(−1)*cos(x)+(0)*sin(x)

4. Simplify the expression by performing the multiplication and addition.

cos(π−x)=−cos(x)+0

cos(π−x)=−cos(x)

Final Answer

cos(π−x)=−cos(x)

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