Let `x=(sum_(n=1)^(44) cos n^(@))/(sum_(n=1)^(44) sin n^(@))` , find the greatest integer that does not exceed.

Answer

To solve the problem, we need to find the value of

\[ x = \frac{\sum_{n=1}^{44} \cos n^\circ}{\sum_{n=1}^{44} \sin n^\circ} \]

### Step 1: Identify the sums
We can express the sums of cosines and sines as follows:

\[ \sum_{n=1}^{44} \cos n^\circ \quad \text{and} \quad \sum_{n=1}^{44} \sin n^\circ \]

### Step 2: Use the formula for the sum of cosines
The sum of cosines can be calculated using the formula:

\[ \sum_{n=1}^{N} \cos n\theta = \frac{\sin\left(\frac{N\theta}{2}\right) \cos\left(\frac{(N+1)\theta}{2}\right)}{\sin\left(\frac{\theta}{2}\right)} \]

For our case, \(N = 44\) and \(\theta = 1^\circ\):

\[ \sum_{n=1}^{44} \cos n^\circ = \frac{\sin\left(\frac{44 \cdot 1^\circ}{2}\right) \cos\left(\frac{(44+1) \cdot 1^\circ}{2}\right)}{\sin\left(\frac{1^\circ}{2}\right)} \]

### Step 3: Calculate the sum of sines
Similarly, the sum of sines can be calculated using the formula:

\[ \sum_{n=1}^{N} \sin n\theta = \frac{\sin\left(\frac{N\theta}{2}\right) \sin\left(\frac{(N+1)\theta}{2}\right)}{\sin\left(\frac{\theta}{2}\right)} \]

For our case:

\[ \sum_{n=1}^{44} \sin n^\circ = \frac{\sin\left(\frac{44 \cdot 1^\circ}{2}\right) \sin\left(\frac{(44+1) \cdot 1^\circ}{2}\right)}{\sin\left(\frac{1^\circ}{2}\right)} \]

### Step 4: Substitute the sums into x
Now substituting the sums into the expression for \(x\):

\[ x = \frac{\frac{\sin(22^\circ) \cos(22.5^\circ)}{\sin(0.5^\circ)}}{\frac{\sin(22^\circ) \sin(22.5^\circ)}{\sin(0.5^\circ)}} \]

### Step 5: Simplify the expression
The \(\sin(0.5^\circ)\) cancels out:

\[ x = \frac{\cos(22.5^\circ)}{\sin(22.5^\circ)} = \cot(22.5^\circ) \]

### Step 6: Calculate \(\cot(22.5^\circ)\)
Using the identity:

\[ \cot(2a) = \frac{\cot^2(a) - 1}{2 \cot(a)} \]

Let \(a = 22.5^\circ\), then \(2a = 45^\circ\) and \(\cot(45^\circ) = 1\):

\[ 1 = \frac{\cot^2(22.5^\circ) - 1}{2 \cot(22.5^\circ)} \]

Let \(y = \cot(22.5^\circ)\):

\[ 1 = \frac{y^2 - 1}{2y} \]

Multiplying through by \(2y\):

\[ 2y = y^2 - 1 \]

Rearranging gives:

\[ y^2 - 2y - 1 = 0 \]

### Step 7: Solve the quadratic equation
Using the quadratic formula:

\[ y = \frac{2 \pm \sqrt{(-2)^2 - 4 \cdot 1 \cdot (-1)}}{2 \cdot 1} = \frac{2 \pm \sqrt{4 + 4}}{2} = \frac{2 \pm 2\sqrt{2}}{2} = 1 \pm \sqrt{2} \]

Since \(\cot(22.5^\circ)\) is positive, we take:

\[ \cot(22.5^\circ) = 1 + \sqrt{2} \]

### Step 8: Find the greatest integer that does not exceed \(x\)
Calculating \(1 + \sqrt{2}\):

\[ \sqrt{2} \approx 1.414 \implies 1 + \sqrt{2} \approx 2.414 \]

Thus, the greatest integer that does not exceed \(x\) is:

\[ \lfloor 2.414 \rfloor = 2 \]

### Final Answer
The greatest integer that does not exceed \(x\) is:

\[ \boxed{2} \]