Expand Using Sum/Difference Formulas (tan(40)+tan(20))/(1-tan(40)tan(20))

Problem

(tan(40)+tan(20))/(1−tan(40)*tan(20))

Solution

1. Identify the structure of the expression as the right-hand side of the tangent sum formula.

2. Recall the tangent sum identity, which states that for any angles A and B

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

3. Match the components of the given expression to the identity by setting A=40 and B=20

4. Apply the formula to condense the expression into a single tangent function.

tan(40+20)

5. Simplify the argument by adding the angles together.

tan(60)

6. Evaluate the trigonometric value using the unit circle or special right triangles.

tan(60)=√(,3)

Final Answer

(tan(40)+tan(20))/(1−tan(40)*tan(20))=√(,3)

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