use a sum or difference identity to find the exact value of the expression.\n\n \tan \frac { 5 pi } { 12 }…

Explanation:

Step1: Express (\frac{5\pi}{12}) as a sum or difference

We know that (\frac{5\pi}{12}=\frac{\pi}{4}+\frac{\pi}{6})

Step2: Use the tangent sum identity

The tangent sum identity is (\tan(A + B)=\frac{\tan A+\tan B}{1-\tan A\tan B}) Here (A = \frac{\pi}{4}), (\tan A=1) and (B=\frac{\pi}{6}), (\tan B=\frac{\sqrt{3}}{3})

Substitute into the formula: [ \begin{align*} \tan\left(\frac{\pi}{4}+\frac{\pi}{6}\right)&=\frac{\tan\frac{\pi}{4}+\tan\frac{\pi}{6}}{1 - \tan\frac{\pi}{4}\tan\frac{\pi}{6}}\ &=\frac{1+\frac{\sqrt{3}}{3}}{1-(1)\times\frac{\sqrt{3}}{3}}\ &=\frac{\frac{3 + \sqrt{3}}{3}}{\frac{3-\sqrt{3}}{3}}\ &=\frac{3+\sqrt{3}}{3-\sqrt{3}} \end{align*} ]

Step3: Rationalize the denominator

Multiply numerator and denominator by (3 + \sqrt{3}) [ \begin{align*} \frac{3+\sqrt{3}}{3-\sqrt{3}}\times\frac{3+\sqrt{3}}{3+\sqrt{3}}&=\frac{(3+\sqrt{3})^2}{9-3}\ &=\frac{9 + 6\sqrt{3}+3}{6}\ &=\frac{12+6\sqrt{3}}{6}\ &=2+\sqrt{3} \end{align*} ]

Answer:

(2+\sqrt{3})