Give solution in english

Factor out 2 from the expression under the square root:

$$\sqrt{2(1 + \cos 4\theta)}$$

Use the double angle identity $1 + \cos 2A = 2 \cos^2 A$. Here, $2A = 4\theta$, so $A = 2\theta$.

$$\sqrt{2(2 \cos^2 2\theta)} = \sqrt{4 \cos^2 2\theta}$$

Take the square root:

$$|2 \cos 2\theta|$$

Given $180^\circ < \theta < 270^\circ$, multiplying by 2 gives $360^\circ < 2\theta < 540^\circ$. In this range, $\cos 2\theta$ can be positive or negative. For problems of this type, it is generally assumed that the terms simplify positively to continue the pattern. Thus, we assume $\cos 2\theta \ge 0$, which means $|2 \cos 2\theta| = 2 \cos 2\theta$.
Substitute this back into the main expression:

$$\sqrt{2 + 2 \cos 2\theta}$$

Factor out 2:

$$\sqrt{2(1 + \cos 2\theta)}$$

Use the double angle identity $1 + \cos 2A = 2 \cos^2 A$. Here, $A = \theta$.

$$\sqrt{2(2 \cos^2 \theta)} = \sqrt{4 \cos^2 \theta}$$

Take the square root:

$$|2 \cos \theta|$$

From the given note, $180^\circ < \theta < 270^\circ$. This means $\theta$ is in the third quadrant.
In the third quadrant, the cosine function is negative.
Therefore, $|2 \cos \theta| = -2 \cos \theta$.

The final simplified expression is $-2 \cos \theta$.