To prove the identity:

$\frac{\tan\left(\frac{A - B}{2}\right)}{\tan\left(\frac{A + B}{2}\right)} = \frac{a - b}{a + b},$

we use the tangent half-angle formulas and properties of trigonometric functions. Let's proceed step by step:

Step 1: Write tangent in terms of sine and cosine

The tangent of an angle can be expressed as: $\tan x = \frac{\sin x}{\cos x}.$

Thus: $\frac{\tan\left(\frac{A - B}{2}\right)}{\tan\left(\frac{A + B}{2}\right)} = \frac{\frac{\sin\left(\frac{A - B}{2}\right)}{\cos\left(\frac{A - B}{2}\right)}}{\frac{\sin\left(\frac{A + B}{2}\right)}{\cos\left(\frac{A + B}{2}\right)}}.$

Simplify: $\frac{\tan\left(\frac{A - B}{2}\right)}{\tan\left(\frac{A + B}{2}\right)} = \frac{\sin\left(\frac{A - B}{2}\right) \cos\left(\frac{A + B}{2}\right)}{\cos\left(\frac{A - B}{2}\right) \sin\left(\frac{A + B}{2}\right)}.$

This becomes: $\frac{\tan\left(\frac{A - B}{2}\right)}{\tan\left(\frac{A + B}{2}\right)} = \frac{\sin\left(\frac{A - B}{2}\right) \cos\left(\frac{A + B}{2}\right) - \cos\left(\frac{A - B}{2}\right) \sin\left(\frac{A + B}{2}\right)}{\cos\left(\frac{A - B}{2}\right) \sin\left(\frac{A + B}{2}\right)}.$

Step 2: Simplify using the sine subtraction formula

The numerator is of the form: $\sin\left(\frac{A - B}{2}\right)\cos\left(\frac{A + B}{2}\right) - \cos\left(\frac{A - B}{2}\right)\sin\left(\frac{A + B}{2}\right).$

This is equivalent to: $\sin\left(\frac{A - B}{2} - \frac{A + B}{2}\right).$

Simplify the argument: $\frac{A - B}{2} - \frac{A + B}{2} = -\frac{B}{2}.$

Thus, the numerator becomes: $\sin\left(-\frac{B}{2}\right).$

Using the odd property of sine ($\sin(-x) = -\sin(x)$): $\sin\left(-\frac{B}{2}\right) = -\sin\left(\frac{B}{2}\right).$

Step 3: Simplify the denominator

The denominator remains: $\cos\left(\frac{A - B}{2}\right)\sin\left(\frac{A + B}{2}\right).$

Step 4: Combine the results

Now the expression is: $\frac{\tan\left(\frac{A - B}{2}\right)}{\tan\left(\frac{A + B}{2}\right)} = \frac{-\sin\left(\frac{B}{2}\right)}{\cos\left(\frac{A - B}{2}\right)\sin\left(\frac{A + B}{2}\right)}.$

Use the identities and context to simplify further based on $a = \tan(A/2)$ and $b = \tan(B/2)$. Would you like to continue with this interpretation?