Find the least degree Polynomial with Integer Coefficients whose one of the roots is $ \cos(12^{\circ})$

My Try: we know that $$\cos(5x)=\cos^5x-10\cos^3x\sin^2x+5\cos x\sin^4x$$ Putting $x=12^{\circ}$ and Converting $\sin$ to $\cos$ we have

$$\frac{1}{2}=x^5-10x^3(1-x^2)+5x(1-2x^2+x^4)$$ $\implies$

$$32x^5-40x^3+10x-1=0$$

Is this the Least degree? Please let me know.