What is the exact value of tan((19pi)/12)?

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Express as Sum/Difference: To find the exact value of tan((19pi)/12), we can express (19pi)/12 as a sum or difference of angles for which we know the exact values of the trigonometric functions. The angles we commonly use are multiples of pi/6, pi/4, and pi/3 because their trigonometric values are well known and can be easily calculated.Apply Angle Sum Identity: We can express (19pi)/12 as the sum of (16pi)/12 and (3pi)/12, which simplifies to (4pi)/3 + pi/4. These two angles are more familiar, and we can use the angle sum identity for tangent to find the exact value of tan((19pi)/12).Calculate Exact Values: The angle sum identity for tangent is tan(A + B) = (tan(A) + tan(B)) / (1 - tan(A)tan(B)). We will use A = (4pi)/3 and B = pi/4 for our calculation.Plug into Identity: First, we find the exact values of tan((4pi)/3) and tan(pi/4). Tan((4pi)/3) is equivalent to tan(pi - (pi/3)), which is the same as -tan(pi/3) because tangent is negative in the third quadrant. The exact value of tan(pi/3) is sqrt(3), so tan((4pi)/3) = -sqrt(3). Tan(pi/4) is 1 because it's the tangent of a 45-degree angle.Simplify Expression: Now we can plug these values into the angle sum identity: tan((19pi)/12) = tan((4pi)/3 + pi/4) = (-sqrt(3) + 1) / (1 - (-sqrt(3) * 1)).Rationalize Denominator: Simplifying the expression, we get tan((19pi)/12) = (-sqrt(3) + 1) / (1 + sqrt(3)). To rationalize the denominator, we multiply the numerator and the denominator by the conjugate of the denominator, which is (1 - sqrt(3)).Expand Numerator: Multiplying the numerator and denominator by the conjugate, we get: tan((19pi)/12) = [(-sqrt(3) + 1)(1 - sqrt(3))] / [(1 + sqrt(3))(1 - sqrt(3))].Expand Denominator: Expanding the numerator, we get: (-sqrt(3) + 1)(1 - sqrt(3)) = -3 - sqrt(3) + sqrt(3) - 1. The sqrt(3) terms cancel out, leaving us with -3 - 1.Final Simplification: Expanding the denominator, we get: (1 + sqrt(3))(1 - sqrt(3)) = 1 - 3, which simplifies to -2.Now we have tan((19pi)/12) = (-4) / (-2), which simplifies to 2.

What is the exact value of $\tan\left(\frac{19\pi}{12}\right)$ ?

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