Find the exact value of each expression.
tan (5π/12)

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Recognize that the angle \( \frac{5\pi}{12} \) radians is not one of the standard angles with known tangent values, so we need to express it as a sum or difference of angles whose tangent values we know. For example, \( \frac{5\pi}{12} = \frac{\pi}{3} + \frac{\pi}{4} \).

Recall the tangent addition formula: \[ \tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B} \]. We will use this formula with \( A = \frac{\pi}{3} \) and \( B = \frac{\pi}{4} \).

Substitute the known tangent values: \( \tan \frac{\pi}{3} = \sqrt{3} \) and \( \tan \frac{\pi}{4} = 1 \) into the formula: \[ \tan \left( \frac{\pi}{3} + \frac{\pi}{4} \right) = \frac{\sqrt{3} + 1}{1 - \sqrt{3} \times 1} \].

Simplify the numerator and denominator separately to get the expression \( \frac{\sqrt{3} + 1}{1 - \sqrt{3}} \).

To find the exact value, rationalize the denominator by multiplying numerator and denominator by the conjugate of the denominator \( (1 + \sqrt{3}) \), then simplify the resulting expression.

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