Let `x=sin1^@` then find the value of the expression `1/(cos0^@cos1^@)+1/(cos1^@cos2^@)+....+1/(cos44^@cos45^@)`

Options

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

Explanation

To solve the expression

\[ \frac{1}{\cos 0^\circ \cos 1^\circ} + \frac{1}{\cos 1^\circ \cos 2^\circ} + \ldots + \frac{1}{\cos 44^\circ \cos 45^\circ}, \]

we start by letting \( x = \sin 1^\circ \).

### Step 1: Rewrite the expression
We can rewrite each term in the expression by multiplying and dividing by \( \sin 1^\circ \):

\[ \frac{1}{\cos k^\circ \cos (k+1)^\circ} = \frac{\sin 1^\circ}{\sin 1^\circ \cos k^\circ \cos (k+1)^\circ}. \]

Thus, the entire expression becomes:

\[ \sum_{k=0}^{44} \frac{\sin 1^\circ}{\cos k^\circ \cos (k+1)^\circ}. \]

### Step 2: Use the sine subtraction formula
Using the identity \( \sin A - \sin B = 2 \cos\left(\frac{A+B}{2}\right) \sin\left(\frac{A-B}{2}\right) \), we can express:

\[ \sin(k+1)^\circ - \sin k^\circ = 2 \cos\left(\frac{(k+1)+k}{2}\right) \sin\left(\frac{(k+1)-k}{2}\right) = 2 \cos\left(k + \frac{1}{2}\right) \sin\left(\frac{1}{2}\right). \]

### Step 3: Substitute back into the expression
Now, we can rewrite the sum as:

\[ \sum_{k=0}^{44} \frac{\sin 1^\circ}{\cos k^\circ \cos (k+1)^\circ} = \sum_{k=0}^{44} \frac{\sin 1^\circ}{\frac{1}{2}(\sin(k+1)^\circ - \sin k^\circ)}. \]

### Step 4: Simplify the expression
This simplifies to:

\[ \sum_{k=0}^{44} \frac{2 \sin 1^\circ}{\sin(k+1)^\circ - \sin k^\circ}. \]

### Step 5: Telescoping series
Notice that this forms a telescoping series. Most terms will cancel out:

\[ 2 \sin 1^\circ \left( \tan 1^\circ + \tan 2^\circ - \tan 1^\circ + \tan 3^\circ - \tan 2^\circ + \ldots + \tan 45^\circ - \tan 44^\circ \right). \]

### Step 6: Final simplification
The only terms that do not cancel are \( \tan 45^\circ \) and \( \tan 0^\circ \):

\[ = 2 \sin 1^\circ \cdot \tan 45^\circ = 2 \sin 1^\circ \cdot 1 = 2 \sin 1^\circ. \]

### Step 7: Substitute back for \( x \)
Since \( x = \sin 1^\circ \), we have:

\[ \frac{1}{\cos 0^\circ \cos 1^\circ} + \frac{1}{\cos 1^\circ \cos 2^\circ} + \ldots + \frac{1}{\cos 44^\circ \cos 45^\circ} = \frac{2}{x}. \]

### Final Answer
Thus, the value of the expression is:

\[ \frac{2}{\sin 1^\circ}. \]