Find the Exact Value (2tan(15))/(1-tan(15)^2)

Problem

(2*tan(15^∘))/(1−tan^2(15^∘))

Solution

1. Identify the trigonometric identity that matches the structure of the expression. The double angle formula for tangent is:

tan(2*θ)=(2*tan(θ))/(1−tan^2(θ))

2. Substitute the given angle θ=15^∘ into the identity to simplify the expression.

(2*tan(15^∘))/(1−tan^2(15^∘))=tan(2⋅15^∘)

3. Multiply the values inside the tangent function.

tan(2⋅15^∘)=tan(30^∘)

4. Evaluate the tangent of 30^∘ using the unit circle or standard trigonometric values.

tan(30^∘)=1/√(,3)

5. Rationalize the denominator by multiplying the numerator and denominator by √(,3)

1/√(,3)=√(,3)/3

Final Answer

(2*tan(15^∘))/(1−tan^2(15^∘))=√(,3)/3

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