cosx=cos x/2-sin x/2)^2-1

To solve the equation \( \cos x = \left(\cos \frac{x}{2} - \sin \frac{x}{2}\right)^2 - 1 \), we'll first simplify the right side of the equation and then isolate \( x \).

Step 1: Expand the Right Side

Start by expanding the square on the right side:

\[ \left(\cos \frac{x}{2} - \sin \frac{x}{2}\right)^2 = \cos^2 \frac{x}{2} - 2 \cos \frac{x}{2} \sin \frac{x}{2} + \sin^2 \frac{x}{2} \]

Using the Pythagorean identity, \( \cos^2 \theta + \sin^2 \theta = 1 \):

\[ \cos^2 \frac{x}{2} + \sin^2 \frac{x}{2} = 1 \]

Thus, we can rewrite:

\[ \left(\cos \frac{x}{2} - \sin \frac{x}{2}\right)^2 = 1 - 2 \cos \frac{x}{2} \sin \frac{x}{2} \]

Recall that \( 2 \cos \frac{x}{2} \sin \frac{x}{2} = \sin x \) (double angle formula). Therefore:

\[ \left(\cos \frac{x}{2} - \sin \frac{x}{2}\right)^2 = 1 - \sin x \]

Now substitute back into the original equation:

\[ \cos x = \left(1 - \sin x\right) - 1 \]

This simplifies to:

\[ \cos x = -\sin x \]

Step 2: Solve the Equation

We can rewrite \( \cos x + \sin x = 0 \):

\[ \cos x = -\sin x \]

Dividing both sides by \( \cos x \) (for \( \cos x \neq 0 \)) yields:

\[ 1 = -\tan x \]

This gives:

\[ \tan x = -1 \]

Step 3: Find General Solutions

The solutions for \( \tan x = -1 \) occur when:

\[ x = \frac{3\pi}{4} + k\pi \quad (k \in \mathbb{Z}) \]

Conclusion

The complete solution set for the equation \( \cos x = \left(\cos \frac{x}{2} - \sin \frac{x}{2}\right)^2 - 1 \) is:

\[ x = \frac{3\pi}{4} + k\pi, \quad k \in \mathbb{Z} \]