## Task A-3.6.

Calculate the product

$$
\left(1-\frac{\cos 61^{\circ}}{\cos 1^{\circ}}\right)\left(1-\frac{\cos 62^{\circ}}{\cos 2^{\circ}}\right) \ldots\left(1-\frac{\cos 119^{\circ}}{\cos 59^{\circ}}\right)
$$
## Solution.

We observe that all factors are of the same form, so we first calculate

$$
\begin{array}{rlrl}
1-\frac{\cos \left(60^{\circ}+k^{\circ}\right)}{\cos k^{\circ}} & =\frac{\cos k^{\circ}-\cos \left(60^{\circ}+k^{\circ}\right)}{\cos k^{\circ}} & & 1 \text{ point} \\
& =\frac{2 \sin 30^{\circ} \sin \left(30^{\circ}+k^{\circ}\right)}{\cos k^{\circ}} & & 2 \text{ points} \\
& =\frac{\sin \left(30^{\circ}+k^{\circ}\right)}{\cos k^{\circ}} & & 1 \text{ point} \\
& =\frac{\cos \left(90^{\circ}-30^{\circ}-k^{\circ}\right)}{\cos k^{\circ}} & & 2 \text{ points} \\
& =\frac{\cos \left(60^{\circ}-k^{\circ}\right)}{\cos k^{\circ}} . & 1 \text{ point}
\end{array}
$$

Therefore,

$$
\prod_{k=1}^{59}\left(1-\frac{\cos \left(60^{\circ}+k^{\circ}\right)}{\cos k^{\circ}}\right)=\frac{\cos 59^{\circ}}{\cos 1^{\circ}} \cdot \frac{\cos 58^{\circ}}{\cos 2^{\circ}} \cdot \ldots \cdot \frac{\cos 1^{\circ}}{\cos 59^{\circ}}=1
$$

Note: A student who performs a procedure analogous to the first part of the solution for only some specific value of $k$ should receive 5 points.