If `theta` lies in the third quadrant , then the expression `sqrt(( 4 sin^(4) theta + sin^(2) 2 theta )) + 4 cos^(2) ((1)/( 4) pi - ( 1)/( 2) theta )` equals 2.

Answer

To solve the expression \( \sqrt{4 \sin^4 \theta + \sin^2 2\theta} + 4 \cos^2 \left(\frac{\pi}{4} - \frac{1}{2} \theta\right) = 2 \) given that \( \theta \) lies in the third quadrant, we will follow these steps:

### Step 1: Simplify \( \sin^2 2\theta \)
We know that:
\[ \sin 2\theta = 2 \sin \theta \cos \theta \]
Thus,
\[ \sin^2 2\theta = (2 \sin \theta \cos \theta)^2 = 4 \sin^2 \theta \cos^2 \theta \]

### Step 2: Substitute \( \sin^2 2\theta \) into the expression
Now, substitute \( \sin^2 2\theta \) into the original expression:
\[ \sqrt{4 \sin^4 \theta + 4 \sin^2 \theta \cos^2 \theta} \]

### Step 3: Factor out the common term
Factor out \( 4 \) from the square root:
\[ \sqrt{4(\sin^4 \theta + \sin^2 \theta \cos^2 \theta)} = 2 \sqrt{\sin^4 \theta + \sin^2 \theta \cos^2 \theta} \]

### Step 4: Simplify \( \sin^4 \theta + \sin^2 \theta \cos^2 \theta \)
Let \( x = \sin^2 \theta \), then \( \cos^2 \theta = 1 - x \):
\[ \sin^4 \theta + \sin^2 \theta \cos^2 \theta = x^2 + x(1-x) = x^2 + x - x^2 = x \]
Thus,
\[ \sqrt{\sin^4 \theta + \sin^2 \theta \cos^2 \theta} = \sqrt{\sin^2 \theta} = |\sin \theta| \]

### Step 5: Consider the sign of \( \sin \theta \)
Since \( \theta \) is in the third quadrant, \( \sin \theta < 0 \). Therefore,
\[ |\sin \theta| = -\sin \theta \]

### Step 6: Substitute back into the expression
Now we have:
\[ 2 \sqrt{\sin^4 \theta + \sin^2 \theta \cos^2 \theta} = 2(-\sin \theta) = -2 \sin \theta \]

### Step 7: Simplify \( 4 \cos^2 \left(\frac{\pi}{4} - \frac{1}{2} \theta\right) \)
Using the cosine subtraction formula:
\[ \cos \left(\frac{\pi}{4} - \frac{1}{2} \theta\right) = \frac{\sqrt{2}}{2} \cos \left(-\frac{1}{2} \theta\right) + \frac{\sqrt{2}}{2} \sin \left(-\frac{1}{2} \theta\right) \]
However, we can also use the identity:
\[ \cos^2 \left(\frac{\pi}{4} - \frac{1}{2} \theta\right) = \frac{1 + \sin \theta}{2} \]
Thus,
\[ 4 \cos^2 \left(\frac{\pi}{4} - \frac{1}{2} \theta\right) = 2(1 + \sin \theta) \]

### Step 8: Combine the results
Now we combine:
\[ -2 \sin \theta + 2(1 + \sin \theta) = -2 \sin \theta + 2 + 2 \sin \theta = 2 \]

### Conclusion
Thus, we have shown that:
\[ \sqrt{4 \sin^4 \theta + \sin^2 2\theta} + 4 \cos^2 \left(\frac{\pi}{4} - \frac{1}{2} \theta\right) = 2 \]