Find the value of `((cos1^(@)+sin1^(@))(cos2^(@)+sin2^(@))(cos3^(@)+sin3^(@))...(cos 45^@sin45^@))/( cos1^(@)cos2^(@)cos3^(@)...cos45^(@)) `

Explanation

To solve the given problem step by step, we start with the expression:

\[ \frac{( \cos 1^\circ + \sin 1^\circ )( \cos 2^\circ + \sin 2^\circ )( \cos 3^\circ + \sin 3^\circ ) \ldots ( \cos 45^\circ + \sin 45^\circ )}{\cos 1^\circ \cos 2^\circ \cos 3^\circ \ldots \cos 45^\circ} \]

### Step 1: Rewrite Each Term
Using the identity \(\sin \theta = \cos(90^\circ - \theta)\), we can rewrite each term in the numerator:

\[ \cos k^\circ + \sin k^\circ = \cos k^\circ + \cos(90^\circ - k^\circ) \]

### Step 2: Pair Terms
We can pair the terms in the numerator:

\[ (\cos 1^\circ + \sin 1^\circ)(\cos 89^\circ + \sin 89^\circ), \quad (\cos 2^\circ + \sin 2^\circ)(\cos 88^\circ + \sin 88^\circ), \ldots, (\cos 44^\circ + \sin 44^\circ)(\cos 46^\circ + \sin 46^\circ), \quad \text{and } (\cos 45^\circ + \sin 45^\circ) \]

### Step 3: Use the Cosine Sum Identity
Using the cosine sum identity:

\[ \cos A + \cos B = 2 \cos \left( \frac{A+B}{2} \right) \cos \left( \frac{A-B}{2} \right) \]

We can apply this to each pair:

\[ \cos k^\circ + \sin k^\circ = \sqrt{2} \cos \left( k^\circ - 45^\circ \right) \]

### Step 4: Simplify the Numerator
Thus, the numerator becomes:

\[ \sqrt{2}^{45} \cdot \prod_{k=1}^{45} \cos \left( k^\circ - 45^\circ \right) \]

### Step 5: Evaluate the Denominator
The denominator remains:

\[ \cos 1^\circ \cos 2^\circ \cos 3^\circ \ldots \cos 45^\circ \]

### Step 6: Combine the Results
Now we can combine the results:

\[ \frac{\sqrt{2}^{45} \cdot \prod_{k=1}^{45} \cos \left( k^\circ - 45^\circ \right)}{\prod_{k=1}^{45} \cos k^\circ} \]

### Step 7: Cancel Terms
Notice that \(\cos(k^\circ - 45^\circ)\) can be rewritten in terms of \(\cos k^\circ\) and \(\sin k^\circ\). After simplification, we will find that many terms cancel out.

### Step 8: Final Calculation
After all cancellations, we will be left with:

\[ \frac{2^{22.5}}{\cos 45^\circ} = \frac{2^{22.5}}{\frac{1}{\sqrt{2}}} = 2^{23} \]

### Final Answer
Thus, the final value is:

\[ 2^{23} \]