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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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Let f: ℝ → ℝ be such that f(tan x + tan y) = f(tan x) + f(tan y) and f(1) = -1. Then the value of the expression (1 - cos(1°)) / Σ [cot(k°) * csc(k°) * cos(k°) + f(k)], where k ranges from 1 to 89, is sought. Multiple-choice options are given.

Explore a trigonometric problem involving a functional equation and summation over angles from 1° to 89°. This problem utilizes key concepts from trigonometry and functional equations to find the value of the given expression. Suitable for advanced students.

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