Verify the Identity tan(2pi-x)=-tan(x)

Problem

tan(2*π−x)=−tan(x)

Solution

1. Identify the tangent subtraction identity, which states that tan(A−B)=(tan(A)−tan(B))/(1+tan(A)*tan(B))

2. Substitute the values A=2*π and B=x into the identity.

tan(2*π−x)=(tan(2*π)−tan(x))/(1+tan(2*π)*tan(x))

3. Evaluate the value of tan(2*π) Since tan(2*π)=0 substitute 0 into the expression.

tan(2*π−x)=(0−tan(x))/(1+(0)*tan(x))

4. Simplify the numerator and the denominator.

tan(2*π−x)=(−tan(x))/1

5. Conclude that the expression simplifies to the right side of the identity.

tan(2*π−x)=−tan(x)

Final Answer

tan(2*π−x)=−tan(x)

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