If π/2 less than x less than π, then write the value of √2 + √2 + 2cos2x in the simplest form.

If π/2 < x < π, Then Find the Simplest Form of √(2 + √(2 + 2cos2x))

If \[ \frac{\pi}{2}

Using the identity

\[ 1+\cos2x=2\cos^2x \]

Therefore,

\[ 2+2\cos2x = 4\cos^2x \] \[ \sqrt{2+2\cos2x} = 2|\cos x| \]

Since

\[ \frac{\pi}{2} \(x\) lies in the second quadrant, where \(\cos x<0\). Hence,

\[ |\cos x| = -\cos x \] \[ \sqrt{2+2\cos2x} = -2\cos x \]

Substituting this value,

\[ \sqrt{\,2+\sqrt{\,2+2\cos2x\,}} = \sqrt{\,2-2\cos x\,} \]

Using the identity

\[ 1-\cos x = 2\sin^2\frac{x}{2} \] \[ 2-2\cos x = 4\sin^2\frac{x}{2} \] \[ \sqrt{\,2-2\cos x\,} = 2\left|\sin\frac{x}{2}\right| \]

Since

\[ \frac{\pi}{4} < \frac{x}{2} < \frac{\pi}{2}, \]

\(\sin\frac{x}{2}\) is positive. Therefore,

\[ 2\left|\sin\frac{x}{2}\right| = 2\sin\frac{x}{2} \]

\[ \boxed{2\sin\frac{x}{2}} \]

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