Questions: Use identities to find the exact value. tan (-5π/12) tan (-5π/12)= (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.)

Solution Steps

To find the exact value of \(\tan \left(-\frac{5 \pi}{12}\right)\), we can use the tangent addition formula. First, express \(-\frac{5 \pi}{12}\) as a sum or difference of angles whose tangent values are known. One possible way is to write it as \(-\frac{5 \pi}{12} = -\frac{\pi}{4} - \frac{\pi}{6}\). Then, apply the tangent addition formula: \(\tan(a - b) = \frac{\tan a - \tan b}{1 + \tan a \tan b}\).

We start by expressing \(-\frac{5 \pi}{12}\) as a sum of angles: \[ -\frac{5 \pi}{12} = -\frac{\pi}{4} - \frac{\pi}{6} \]

Next, we calculate the tangent values of the angles: \[ \tan\left(-\frac{\pi}{4}\right) = -1 \] \[ \tan\left(-\frac{\pi}{6}\right) = -\frac{\sqrt{3}}{3} \]

Using the tangent addition formula: \[ \tan(a - b) = \frac{\tan a - \tan b}{1 + \tan a \tan b} \] we substitute \(a = -\frac{\pi}{4}\) and \(b = -\frac{\pi}{6}\): \[ \tan\left(-\frac{5 \pi}{12}\right) = \frac{-1 - \left(-\frac{\sqrt{3}}{3}\right)}{1 + \left(-1\right)\left(-\frac{\sqrt{3}}{3}\right)} \]

This simplifies to: \[ \tan\left(-\frac{5 \pi}{12}\right) = \frac{-1 + \frac{\sqrt{3}}{3}}{1 + \frac{\sqrt{3}}{3}} = \frac{-\frac{3}{3} + \frac{\sqrt{3}}{3}}{\frac{3}{3} + \frac{\sqrt{3}}{3}} = \frac{-3 + \sqrt{3}}{3 + \sqrt{3}} \]

Further simplifying gives: \[ \tan\left(-\frac{5 \pi}{12}\right) = -2 + \sqrt{3} \]

Final Answer