The image contains the following mathematical question:

Let $f : \mathbb{R} \to \mathbb{R}$ be such that: $f(\text{tan }x + \text{tan }y) = f(\text{tan }x) + f(\text{tan }y) \quad \text{and} \quad f(1) = -1$ Then the value of: $\frac{1 - \cos 1^\circ}{\sum_{k=1}^{89} \cot k^\circ \cdot \csc k^\circ \cdot \cos k^\circ + f(k)}$

The multiple-choice options are:

1. $\frac{\sin 20^\circ}{\sin 19^\circ}$
2. $\frac{\cos 20^\circ}{\cos 19^\circ}$
3. $\frac{\sin 20^\circ}{\sin 21^\circ}$
4. $\frac{\cos 20^\circ}{\sin 20^\circ}$
5. 0

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