If `z_(1) and z_(2)` are two nonzero complex numbers and `theta` is a real number, then `(1)/(|z_(1)|^(2) + |z_(2)|^(2))[|(cos theta)z_(1) - (sin theta)z_(2)|^(2) + |(sin theta)z_(1) + (cos theta) z_(2)|^(2) ]` is equal to ________

Answer

To solve the given expression

\[ \frac{1}{|z_1|^2 + |z_2|^2} \left[ |(\cos \theta) z_1 - (\sin \theta) z_2|^2 + |(\sin \theta) z_1 + (\cos \theta) z_2|^2 \right], \]

we will follow these steps:

### Step 1: Expand the terms inside the absolute value
We start with the first term \( |(\cos \theta) z_1 - (\sin \theta) z_2|^2 \):

\[ |(\cos \theta) z_1 - (\sin \theta) z_2|^2 = ((\cos \theta) z_1 - (\sin \theta) z_2)((\cos \theta) z_1 - (\sin \theta) z_2)^*, \]

where \(*\) denotes the complex conjugate.

### Step 2: Compute the conjugate
The conjugate of \( (\cos \theta) z_1 - (\sin \theta) z_2 \) is \( (\cos \theta) z_1^* - (\sin \theta) z_2^* \). Thus, we have:

\[ |(\cos \theta) z_1 - (\sin \theta) z_2|^2 = (\cos^2 \theta |z_1|^2 - \cos \theta \sin \theta (z_1 z_2^* + z_2 z_1^*) + \sin^2 \theta |z_2|^2). \]

### Step 3: Expand the second term
Now we compute the second term \( |(\sin \theta) z_1 + (\cos \theta) z_2|^2 \):

\[ |(\sin \theta) z_1 + (\cos \theta) z_2|^2 = ((\sin \theta) z_1 + (\cos \theta) z_2)((\sin \theta) z_1 + (\cos \theta) z_2)^*. \]

The conjugate is \( (\sin \theta) z_1^* + (\cos \theta) z_2^* \). Thus, we have:

\[ |(\sin \theta) z_1 + (\cos \theta) z_2|^2 = (\sin^2 \theta |z_1|^2 + \sin \theta \cos \theta (z_1 z_2^* + z_2 z_1^* ) + \cos^2 \theta |z_2|^2). \]

### Step 4: Combine the two expanded terms
Now we combine both expansions:

\[ |(\cos \theta) z_1 - (\sin \theta) z_2|^2 + |(\sin \theta) z_1 + (\cos \theta) z_2|^2 = (\cos^2 \theta |z_1|^2 + \sin^2 \theta |z_1|^2) + (\sin^2 \theta |z_2|^2 + \cos^2 \theta |z_2|^2). \]

### Step 5: Simplify the expression
Using the identity \( \cos^2 \theta + \sin^2 \theta = 1 \):

\[ = |z_1|^2 (\cos^2 \theta + \sin^2 \theta) + |z_2|^2 (\sin^2 \theta + \cos^2 \theta) = |z_1|^2 + |z_2|^2. \]

### Step 6: Substitute back into the original expression
Now substituting back into the original expression:

\[ \frac{1}{|z_1|^2 + |z_2|^2} (|z_1|^2 + |z_2|^2) = \frac{|z_1|^2 + |z_2|^2}{|z_1|^2 + |z_2|^2} = 1. \]

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

\[ \boxed{1}. \]