To find $\tan B$ in terms of $\tan A$, given that $\tan A = \frac{\cos(A+B)}{\cos(A-B)}$, we can use trigonometric identities and relationships.

Given:

$\tan A = \frac{\cos(A + B)}{\cos(A - B)}$

We will use the sum and difference formulas for cosine:

$\cos(A + B) = \cos A \cos B - \sin A \sin B$

$\cos(A - B) = \cos A \cos B + \sin A \sin B$

Substitute these into the equation for $\tan A$:

$\tan A = \frac{\cos A \cos B - \sin A \sin B}{\cos A \cos B + \sin A \sin B}$

Divide both the numerator and the denominator by $\cos A \cos B$:

$\tan A = \frac{\frac{\cos A \cos B - \sin A \sin B}{\cos A \cos B}}{\frac{\cos A \cos B + \sin A \sin B}{\cos A \cos B}}$

Simplify by separating the fractions:

$\tan A = \frac{1 - \frac{\sin A \sin B}{\cos A \cos B}}{1 + \frac{\sin A \sin B}{\cos A \cos B}}$

We can use the identity $\sin X = \tan X \cos X$ to rewrite the fractions:

$\tan A = \frac{1 - \frac{\tan A \tan B}{\cos A \cos B}}{1 + \frac{\tan A \tan B}{\cos A \cos B}}$

Simplify further by observing that $\frac{\tan A \tan B}{\cos A \cos B} = \tan A \tan B$:

$\tan A = \frac{1 - \tan A \tan B}{1 + \tan A \tan B}$

Rearrange this equation to isolate $\tan B$:

$\tan A (1 + \tan A \tan B) = 1 - \tan A \tan B$

Expand and rearrange to get terms involving $\tan B$ on one side:

$\tan A + \tan A^2 \tan B = 1 - \tan A \tan B$

$\tan A^2 \tan B + \tan A \tan B = 1 - \tan A$

Factor out $\tan B$ from the left-hand side:

$\tan B (\tan A^2 + \tan A) = 1 - \tan A$

Solve for $\tan B$:

$\tan B = \frac{1 - \tan A}{\tan A^2 + \tan A}$

Thus, $\tan B$ in terms of $\tan A$ is:

$\tan B = \frac{1 - \tan A}{\tan A^2 + \tan A}$

Would you like more details or have any questions about this solution?

Here are 5 related questions to expand your understanding:

1. What is the value of $\tan B$ if $\tan A = 1$?
2. How would you find $\tan(A + B)$ in terms of $\tan A$ and $\tan B$?
3. Can you derive an expression for $\sin B$ in terms of $\sin A$ and $\cos A$?
4. How do the sum and difference formulas for sine and cosine relate to the tangent formula used here?
5. What happens to the value of $\tan B$ as $\tan A$ approaches infinity?

Tip: To simplify trigonometric expressions, always consider using fundamental identities like the Pythagorean identities and sum/difference formulas.