Evaluate

Problem

(tan(π/18)+tan(π/9))/(1−tan(π/18)*tan(π/9))

Solution

1. Identify the trigonometric identity for the tangent of a sum of two angles.

tan(A+B)=(tan(A)+tan(B))/(1−tan(A)*tan(B))

2. Match the given expression to the identity by setting the angle values.

A=π/18

B=π/9

3. Apply the sum formula to rewrite the expression as a single tangent function.

tan(π/18+π/9)

4. Find a common denominator to add the fractions inside the tangent function.

π/18+(2*π)/18=(3*π)/18

5. Simplify the resulting fraction.

(3*π)/18=π/6

6. Evaluate the tangent of the simplified angle using standard trigonometric values.

tan(π/6)=1/√(,3)

7. Rationalize the denominator if necessary.

1/√(,3)=√(,3)/3

Final Answer

(tan(π/18)+tan(π/9))/(1−tan(π/18)*tan(π/9))=√(,3)/3

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