simplify this expression $1+\cos x+\sqrt{\frac{1+\cos x}{2}}$ $$1+\cos x+\sqrt{\frac{1+\cos x}{2}}=0$$
Can someone help me to solve 
I tried with a quadratic equation but I can't get the right result.
 A: $$1+\cos x+\sqrt{\frac{1+\cos x}{2}}=0\implies 1+\cos x=2+4\cos x+2\cos^2x\implies$$
$$2\cos^2x+3\cos x+1=0$$
The above quadratic (in cosine) discriminant is
$$\Delta=9-8=1$$
so you get
$$\cos x_{1,2}=\frac{-3\pm 1}{4}=\begin{cases}-1\\{}\\-\frac12\end{cases}$$
Solve thus $\;\cos x=-1\;,\;\;\cos x=-\frac12\;$ ...and remember afterwards to check the possible solutions in the original equation since we squared and things can go wrong with this (why?)
A: Note that: $$\sqrt{\frac{1 + \cos x}{2}} = \cos \frac x2; \quad \text{for}  \quad  \cos \frac x2 \ge 0$$
From this we have: $1 + \cos x = 2 \cos^2 \frac x2$
Now just substitute and we have:
$$2\cos^2 \frac x2 + \cos \frac x2 = 0$$
$$\cos \frac x2 \left(2\cos \frac x2 + 1\right) = 0$$
Now solve it.
A: For your problem to have a solution other than $1+\cos(x)=0$ you need to consider the "other" square root, viz, the negative square root. In this case
$$
1+\cos(x) = \frac 1 2
$$
works if you are willing to fudge the definition of square roots. If not there is no solution other than $1+\cos(x)=0$
