Half angle identities

The trigonometric half-angle identities state the following equalities:

  • $\sin \left(\frac{x}{2}\right) = \pm \sqrt{\frac{1 - \cos (x)}{2}}$
  • $\cos \left(\frac{x}{2}\right) = \pm \sqrt{\frac{1 + \cos (x)}{2}}$
  • $\tan \left(\frac{x}{2}\right) = \pm \sqrt{\frac{1 - \cos (x)}{1+\cos (x)}} = \frac{\sin (x)}{1 + \cos (x)} = \frac{1-\cos (x)}{\sin (x)}$

The plus or minus does not mean that there are two answers, but that the sign of the expression depends on the quadrant in which the angle resides.

Consider the two expressions listed in the cosine double-angle section for $\sin^2 (x)$ and $\cos^2 (x)$, and substitute $\frac{1}{2} x$ instead of $x$. Taking the square root then yields the desired half-angle identities for sine and cosine. As for the tangent identity, divide the sine and cosine half-angle identities.

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