https://file.aiquickdraw.com/mathbot/file/1725456267005mfpvxhls.jpg

To solve \(\tan(105^\circ)\) exactly, we can break down \(105^\circ\) as a sum of angles for which the tangent values are known. We can use the angle sum identity for tangent:

\[
\tan(A + B) = \frac{\tan(A) + \tan(B)}{1 - \tan(A)\tan(B)}
\]

Let’s express \(105^\circ\) as \(60^\circ + 45^\circ\). Then:

\[
\tan(105^\circ) = \tan(60^\circ + 45^\circ)
\]

Using the identity:

\[
\tan(105^\circ) = \frac{\tan(60^\circ) + \tan(45^\circ)}{1 - \tan(60^\circ)\tan(45^\circ)}
\]

We know:

\[
\tan(60^\circ) = \sqrt{3} \quad \text{and} \quad \tan(45^\circ) = 1
\]

Substituting these values in:

\[
\tan(105^\circ) = \frac{\sqrt{3} + 1}{1 - \sqrt{3} \cdot 1} = \frac{\sqrt{3} + 1}{1 - \sqrt{3}}
\]

To simplify \(\frac{\sqrt{3} + 1}{1 - \sqrt{3}}\), multiply the numerator and the denominator by the conjugate of the denominator:

\[
\tan(105^\circ) = \frac{(\sqrt{3} + 1)(1 + \sqrt{3})}{(1 - \sqrt{3})(1 + \sqrt{3})}
\]

Simplifying the denominator:

\[
(1 - \sqrt{3})(1 + \sqrt{3}) = 1 - 3 = -2
\]

Now, expand the numerator:

\[
(\sqrt{3} + 1)(1 + \sqrt{3}) = \sqrt{3} + 3 + 1 + \sqrt{3} = 4 + 2\sqrt{3}
\]

So:

\[
\tan(105^\circ) = \frac{4 + 2\sqrt{3}}{-2} = -2 - \sqrt{3}
\]

Therefore, the exact value of \(\tan(105^\circ)\) is \(-2 - \sqrt{3}\). 

Looking at the options given in the image, the correct choice is:

\[
\boxed{-2 - \sqrt{3}}
\]

Would you like more details on the process, or do you have any other questions? Here are some related questions:

1. How do you compute the exact value of \(\tan(75^\circ)\)?
2. What are the other angle sum identities for sine and cosine?
3. How can you derive the value of \(\sin(105^\circ)\) using angle addition formulas?
4. What is the geometric interpretation of the tangent function?
5. How do you simplify complex expressions involving radicals?

**Tip:** When simplifying trigonometric expressions involving radicals, multiplying by the conjugate can often simplify the problem significantly.

Learn how to compute the exact value of tan(105°) using the angle addition formula. This problem covers essential trigonometric concepts and is suitable for students in Grades 10-12.

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