Verify the Identity tan(3pi+x)=tan(x)

Problem

tan(3*π+x)=tan(x)

Solution

1. Identify the periodicity of the tangent function. The tangent function tan(θ) has a period of π which means tan(θ+n*π)=tan(θ) for any integer n

2. Apply the periodicity property to the given expression where n=3 Since 3 is an integer, the value of the tangent function remains unchanged when 3*π is added to the angle.

3. Substitute the property into the expression to show that the left side equals the right side.

4. Alternatively, use the sum formula for tangent: tan(A+B)=(tan(A)+tan(B))/(1−tan(A)*tan(B))

5. Evaluate the sum formula with A=3*π and B=x Since tan(3*π)=0 the expression simplifies directly.

tan(3*π+x)=(tan(3*π)+tan(x))/(1−tan(3*π)*tan(x))

tan(3*π+x)=(0+tan(x))/(1−(0)*tan(x))

tan(3*π+x)=tan(x)/1

Final Answer

tan(3*π+x)=tan(x)

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