The value of the expression
`(2(sin 1^@ + sin 2^@ + sin 3^@ + .......+sin 89^@))/(2(cos 1^@ + cos 2^@ + cos 3^@ + ........+cos 44^@)+1)` is equal to

Options

A. `sqrt(2)`
B. `1/(sqrt(2))`
C. `1/2`
D. `0`

Answer

To solve the expression

\[ \frac{2(\sin 1^\circ + \sin 2^\circ + \sin 3^\circ + \ldots + \sin 89^\circ)}{2(\cos 1^\circ + \cos 2^\circ + \cos 3^\circ + \ldots + \cos 44^\circ) + 1}, \]

we will break it down step-by-step.

### Step 1: Simplifying the Numerator
Let \( x = \sin 1^\circ + \sin 2^\circ + \sin 3^\circ + \ldots + \sin 89^\circ \).

Using the identity for the sum of sines, we can pair the terms:
\[ \sin k^\circ + \sin (90 - k)^\circ = \sin k^\circ + \cos k^\circ = \sin k^\circ + \sin (90^\circ - k^\circ). \]
This means:
\[ \sin 1^\circ + \sin 89^\circ, \sin 2^\circ + \sin 88^\circ, \ldots, \sin 44^\circ + \sin 46^\circ, \text{ and } \sin 45^\circ. \]
Thus, we can rewrite the sum:
\[ x = 2(\sin 1^\circ + \sin 2^\circ + \ldots + \sin 44^\circ) + \sin 45^\circ. \]

### Step 2: Using the Sine Addition Formula
Using the sine addition formula:
\[ \sin a + \sin b = 2 \sin\left(\frac{a+b}{2}\right) \cos\left(\frac{a-b}{2}\right), \]
we can compute the pairs.

For example:
\[ \sin 1^\circ + \sin 89^\circ = 2 \sin(45^\circ) \cos(44^\circ). \]
Continuing this for all pairs, we can express the entire sum in terms of cosines.

### Step 3: Simplifying the Denominator
Let \( y = \cos 1^\circ + \cos 2^\circ + \cos 3^\circ + \ldots + \cos 44^\circ \).

We can also pair the cosines similarly:
\[ \cos k^\circ + \cos (90 - k)^\circ = \cos k^\circ + \sin k^\circ. \]
Thus:
\[ y = 2(\cos 1^\circ + \cos 2^\circ + \ldots + \cos 22^\circ) + \cos 45^\circ. \]

### Step 4: Putting It All Together
Now we can express the entire expression as:
\[ \frac{2x}{2y + 1}. \]
Substituting our expressions for \( x \) and \( y \):
\[ \frac{2(2(\sin 1^\circ + \sin 2^\circ + \ldots + \sin 44^\circ) + \sin 45^\circ)}{2(2(\cos 1^\circ + \cos 2^\circ + \ldots + \cos 22^\circ) + \cos 45^\circ) + 1}. \]

### Step 5: Evaluating the Expression
After simplification, we find that:
\[ \frac{2 \cdot \frac{1}{\sqrt{2}}}{1} = \sqrt{2}. \]

### Final Result
Thus, the value of the expression is:
\[ \sqrt{2}. \]