To find the exact value of [tex]\(\tan 75^\circ\)[/tex], we can use the tangent addition formula. Specifically, we know that:
[tex]\[ 75^\circ = 45^\circ + 30^\circ \][/tex]
The tangent addition formula is:
[tex]\[ \tan(a + b) = \frac{\tan a + \tan b}{1 - \tan a \tan b} \][/tex]
Applying this to [tex]\(a = 45^\circ\)[/tex] and [tex]\(b = 30^\circ\)[/tex]:
1. We know that:
[tex]\[ \tan 45^\circ = 1 \][/tex]
[tex]\[ \tan 30^\circ = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3} \][/tex]
2. Now, substitute these values into the tangent addition formula:
[tex]\[ \tan 75^\circ = \frac{\tan 45^\circ + \tan 30^\circ}{1 - \tan 45^\circ \tan 30^\circ} \][/tex]
[tex]\[ \tan 75^\circ = \frac{1 + \frac{\sqrt{3}}{3}}{1 - 1 \cdot \frac{\sqrt{3}}{3}} \][/tex]
3. Simplify the expression:
[tex]\[ \tan 75^\circ = \frac{1 + \frac{\sqrt{3}}{3}}{1 - \frac{\sqrt{3}}{3}} \][/tex]
Thus, the correct answer is:
[tex]\[ \boxed{\frac{1+\frac{\sqrt{3}}{3}}{1-\frac{\sqrt{3}}{3}}} \][/tex]
The corresponding option in the given choices is:
B. [tex]\(\frac{1+\frac{\sqrt{3}}{3}}{1-\frac{\sqrt{3}}{3}}\)[/tex]
