### Instruction: Let $x=\frac{\sum\limits_{n=1}^{44} \cos n^\circ}{\sum\limits_{n=1}^{44} \sin n^\circ}$. What is the greatest integer that does not exceed $100x$?### Response: I notice that the numerator and denominator of $x$ are both sums of trigonometric functions with angles in degrees.
I recall that there is a formula for the sum of cosine or sine of angles in arithmetic progression, namely
$$\sum\limits_{k=0}^{n-1} \cos (a + kd) = \frac{\sin \frac{nd}{2}}{\sin \frac{d}{2}} \cos \left(a + \frac{(n-1)d}{2}\right)$$
and
$$\sum\limits_{k=0}^{n-1} \sin (a + kd) = \frac{\sin \frac{nd}{2}}{\sin \frac{d}{2}} \sin \left(a + \frac{(n-1)d}{2}\right)$$
I apply this formula to the numerator and denominator of $x$, with $a=1^\circ$, $d=1^\circ$, and $n=44$.
I get
$$x = \frac{\frac{\sin 22^\circ}{\sin \frac{1}{2}^\circ} \cos 22^\circ}{\frac{\sin 22^\circ}{\sin \frac{1}{2}^\circ} \sin 22^\circ} = \frac{\cos 22^\circ}{\sin 22^\circ} = \cot 22^\circ$$
I use a calculator to find the value of $\cot 22^\circ$ as a decimal approximation.
I get $\cot 22^\circ \approx 2.47509$.
I multiply this value by 100 to get $100x \approx 247.509$.
I round this value down to the nearest integer, since the problem asks for the greatest integer that does not exceed $100x$.
I get 247 as the final answer.