Sine rule, cosine rule, but tangent rule?

Geometry Level 4

\[ \frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C} \\ \text{ } \\ \text{and} \\ \text{ } \\ a ^2 = b^2 + c^2 - 2bc\cos A. \]

I think most of us here know the sine rule and cosine rule, which, in case you don't know, are as shown above.

But, did you know there exists a tangent rule as well? Here it is!

Express \( \dfrac{\tan \frac{A+B}{2}}{\tan \frac{A-B}{2}}\) in terms of \(a\) and \(b\).

Note: Assume all denominators in the fractions in all the answer choices are non-zero.


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