To find the exact value of [tex]\(\tan \frac{7\pi}{12}\)[/tex], we can use angle sum identities.
First, note that:
[tex]\[ \frac{7\pi}{12} = \frac{3\pi}{12} + \frac{4\pi}{12} = \frac{\pi}{4} + \frac{\pi}{3} \][/tex]
We use the tangent angle sum identity:
[tex]\[ \tan (A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B} \][/tex]
Set [tex]\( A = \frac{\pi}{4} \)[/tex] and [tex]\( B = \frac{\pi}{3} \)[/tex].
Now, we know:
[tex]\[ \tan\left(\frac{\pi}{4}\right) = 1 \][/tex]
[tex]\[ \tan\left(\frac{\pi}{3}\right) = \sqrt{3} \][/tex]
Using the tangent sum identity:
[tex]\[ \tan \left( \frac{7\pi}{12} \right) = \tan \left( \frac{\pi}{4} + \frac{\pi}{3} \right) \][/tex]
[tex]\[ \tan \left( \frac{\pi}{4} + \frac{\pi}{3} \right) = \frac{\tan \left( \frac{\pi}{4} \right) + \tan \left( \frac{\pi}{3} \right)}{1 - \tan \left( \frac{\pi}{4} \right) \tan \left( \frac{\pi}{3} \right)} \][/tex]
Substitute the known values:
[tex]\[ \tan \left( \frac{\pi}{4} + \frac{\pi}{3} \right) = \frac{1 + \sqrt{3}}{1 - 1 \cdot \sqrt{3}} \][/tex]
Solve the denominator:
[tex]\[ 1 - \sqrt{3} \][/tex]
So:
[tex]\[ \tan \left( \frac{\pi}{4} + \frac{\pi}{3} \right) = \frac{1 + \sqrt{3}}{1 - \sqrt{3}} \][/tex]
To simplify this, multiply the numerator and the denominator by the conjugate of the denominator [tex]\( 1 + \sqrt{3} \)[/tex]:
[tex]\[ \frac{(1 + \sqrt{3})(1 + \sqrt{3})}{(1 - \sqrt{3})(1 + \sqrt{3})} \][/tex]
Expand the numerator:
[tex]\[ (1 + \sqrt{3})(1 + \sqrt{3}) = 1 + 2\sqrt{3} + 3 = 4 + 2\sqrt{3} \][/tex]
Expand the denominator using the difference of squares:
[tex]\[ (1 - \sqrt{3})(1 + \sqrt{3}) = 1 - (\sqrt{3})^2 = 1 - 3 = -2 \][/tex]
So, the tangent value becomes:
[tex]\[ \frac{4 + 2\sqrt{3}}{-2} = -2 - \sqrt{3} \][/tex]
This simplified form confirms that the exact value is:
[tex]\[ \tan \left( \frac{7\pi}{12} \right) = -2 - \sqrt{3} \][/tex]
Thus, the correct option that matches this value is:
[tex]\[ \boxed{\text{c) } -\frac{\sqrt{2-\sqrt{3}}}{\sqrt{2+\sqrt{3}}}} \][/tex]
However, I must correct myself. The derivation using the conjugate multiplication seems to have omitted a step where the radical simplification leads to another format, making sure it conforms exactly to the given choices. The correct matching form among the choices is:
[tex]\[ \boxed{ -\frac{\sqrt{2+\sqrt{3}}}{\sqrt{2-\sqrt{3}}}} \][/tex]
Thus, the suitable correct option based on value conformance is indeed option (b).
