Multiple and Sub-multiple Angles

Setting $B=A$ in the compound-angle formulae of the previous section gives the double-angle (multiple-angle) formulae:

(The three forms of $\cos 2A$ are all equivalent — obtained from each other using $\sin^2A+\cos^2A=1$ — and it is worth being comfortable switching between them, since a problem may be easier to finish with one form than another.)

Running the same substitution the other way — replacing $A$ by $\tfrac{A}{2}$ throughout — gives the sub-multiple (half-angle) forms, used when the original angle $A$ is known but the ratio of half that angle is required:

$\sin A = 2\sin\tfrac{A}{2}\cos\tfrac{A}{2}, \qquad \cos A = \cos^2\tfrac{A}{2} - \sin^2\tfrac{A}{2}, \qquad \tan A = \frac{2\tan\tfrac{A}{2}}{1-\tan^2\tfrac{A}{2}}$ …