Find the value of `(sin^2 63^@ + sin^2 27^@)/(cos^2 17 + cos^2 73^@)`.

Answer

To solve the expression \((\sin^2 63^\circ + \sin^2 27^\circ) / (\cos^2 17^\circ + \cos^2 73^\circ)\), we can follow these steps:

### Step 1: Simplifying the numerator
We start with the numerator:
\[ \sin^2 63^\circ + \sin^2 27^\circ \]
Using the identity \(\sin^2 \theta + \sin^2 (90^\circ - \theta) = 1\), we can rewrite \(\sin^2 27^\circ\) as:
\[ \sin^2 27^\circ = \cos^2 (90^\circ - 27^\circ) = \cos^2 63^\circ \]
Thus, we have:
\[ \sin^2 63^\circ + \sin^2 27^\circ = \sin^2 63^\circ + \cos^2 63^\circ = 1 \]

### Step 2: Simplifying the denominator
Next, we simplify the denominator:
\[ \cos^2 17^\circ + \cos^2 73^\circ \]
Using the identity \(\cos^2 \theta + \cos^2 (90^\circ - \theta) = 1\), we can rewrite \(\cos^2 73^\circ\) as:
\[ \cos^2 73^\circ = \sin^2 (90^\circ - 73^\circ) = \sin^2 17^\circ \]
Thus, we have:
\[ \cos^2 17^\circ + \cos^2 73^\circ = \cos^2 17^\circ + \sin^2 17^\circ = 1 \]

### Step 3: Putting it all together
Now we can substitute back into our original expression:
\[ \frac{\sin^2 63^\circ + \sin^2 27^\circ}{\cos^2 17^\circ + \cos^2 73^\circ} = \frac{1}{1} = 1 \]

### Final Answer
The value of the expression is:
\[ \boxed{1} \]