Can I input negative angles into the cosine half-angle formula?

Question

So the cosine half-angle formula says:

Now, we know that co-terminal angles have equal cosines. Consider that $\cos (7\pi/4)$ = $\cos(-\pi/4)$. However, if you apply the half angle formula to $(7\pi/4)$ you get a different answer than if you apply the half angle formula to $(-\pi/4)$. Does the half angle formula require that your angle being inputted be positive, so that applying the half-angle formula to $(-\pi/4)$ would mean plugging in the positive coterminal version, $(7\pi/4)$, to the formula? (In which case the contradiction disappears)

Answer 1

For any angle $\theta$, we have$$\frac{1+\cos\theta}2=\frac{1+\cos\left(2\frac\theta2\right)}2=\frac{1+\cos^2\left(\frac\theta2\right)-\sin^2\left(\frac\theta2\right)}2=\cos^2\left(\frac\theta2\right),$$since $1-\sin^2\left(\frac\theta2\right)=\cos^2\left(\frac\theta2\right).$ So, yes, that formula is always valid.

Answer 2

Yes, but you need to be mindful of the resulting sign of the answer. If that negative angle ends up in the first or 4th quadrant use the positive sign, if it is in the 2nd or 3rd use negative. This formula is very delicate, if you use the following property of cosines: $\cos(-x)=cos(x)$, then you will realize that the same analysis is required for the positive values.

Simply, be careful of the sign of the cos of your angle and ignore the negative sign in front of the angle.