The expression `(1+sin22^@sin33^@sin35^@)/(cos^2 22^@+cos^2 33^@+cos^2 35^@)` simplifies to

Explanation

To simplify the expression \(\frac{1 + \sin 22^\circ + \sin 33^\circ + \sin 35^\circ}{\cos^2 22^\circ + \cos^2 33^\circ + \cos^2 35^\circ}\), we can follow these steps:

### Step 1: Identify the angles
Let:
- \(a = 22^\circ\)
- \(b = 33^\circ\)
- \(c = 35^\circ\)

### Step 2: Recognize the sum of angles
Notice that:
\[ a + b + c = 22^\circ + 33^\circ + 35^\circ = 90^\circ \]

### Step 3: Use the identity for cosines
We can use the identity for the cosine of angles in a triangle:
\[ \cos^2 a + \cos^2 b + \cos^2 c = 2 + 2 \sin a \sin b \sin c \]
However, since we have \(a + b + c = 90^\circ\), we can use the specific property for angles that sum to \(90^\circ\):
\[ \cos^2 a + \cos^2 b + \cos^2 c = 1 + \sin a \sin b + \sin b \sin c + \sin c \sin a \]

### Step 4: Simplify the numerator
The numerator can be rewritten as:
\[ 1 + \sin 22^\circ + \sin 33^\circ + \sin 35^\circ \]

### Step 5: Multiply numerator and denominator by 2
To simplify the expression, we multiply both the numerator and denominator by 2:
\[ \frac{2(1 + \sin 22^\circ + \sin 33^\circ + \sin 35^\circ)}{2(\cos^2 22^\circ + \cos^2 33^\circ + \cos^2 35^\circ)} \]

### Step 6: Evaluate the denominator using the identity
Using the property mentioned earlier:
\[ \cos^2 22^\circ + \cos^2 33^\circ + \cos^2 35^\circ = 2 + 2 \sin 22^\circ \sin 33^\circ \sin 35^\circ \]

### Step 7: Substitute back into the expression
Now we can substitute this back into our expression:
\[ \frac{2(1 + \sin 22^\circ + \sin 33^\circ + \sin 35^\circ)}{2 + 2 \sin 22^\circ \sin 33^\circ \sin 35^\circ} \]

### Step 8: Simplify the expression
This simplifies to:
\[ \frac{1 + \sin 22^\circ + \sin 33^\circ + \sin 35^\circ}{1 + \sin 22^\circ \sin 33^\circ \sin 35^\circ} \]

### Step 9: Final simplification
Since the numerator and denominator are both multiplied by 2, we can cancel out the 2:
\[ = \frac{1}{1} = 1 \]

Thus, the final simplified expression is:
\[ \frac{1 + \sin 22^\circ + \sin 33^\circ + \sin 35^\circ}{\cos^2 22^\circ + \cos^2 33^\circ + \cos^2 35^\circ} = \frac{1}{2} \]