use a sum or difference identity to find the exact value of the expression.\n\\( \\tan 75 ^ { \\circ }…

Explanation:

Step1: Express (75^{\circ}) as a sum

(75^{\circ}=45^{\circ} + 30^{\circ})

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 = 45^{\circ}) and (B=30^{\circ}), (\tan45^{\circ}=1) and (\tan30^{\circ}=\frac{\sqrt{3}}{3}) [ \begin{align*} \tan75^{\circ}&=\tan(45^{\circ}+ 30^{\circ})\ &=\frac{\tan45^{\circ}+\tan30^{\circ}}{1-\tan45^{\circ}\tan30^{\circ}}\ &=\frac{1+\frac{\sqrt{3}}{3}}{1 - 1\times\frac{\sqrt{3}}{3}} \end{align*} ]

Step3: Simplify the fraction

Multiply the numerator and denominator by (3) to get rid of the fractions: [ \begin{align*} \frac{1+\frac{\sqrt{3}}{3}}{1-\frac{\sqrt{3}}{3}}&=\frac{3 + \sqrt{3}}{3-\sqrt{3}}\ &=\frac{(3+\sqrt{3})(3 + \sqrt{3})}{(3-\sqrt{3})(3+\sqrt{3})}\ &=\frac{9+6\sqrt{3}+3}{9 - 3}\ &=\frac{12 + 6\sqrt{3}}{6}\ &=2+\sqrt{3} \end{align*} ]

Answer:

(2+\sqrt{3})