Video narration
How can one angle give you the sine, cosine, and tangent of twice itself? Make two identical turns with a protractor: the total is theta plus theta, or two theta. Doubling is not a new trick; it is angle addition with the same angle twice. For sine, substitute theta twice: sine of two theta equals sine theta cosine theta plus cosine theta sine theta, giving two sine theta cosine theta. For cosine, substitution gives cosine of two theta equals cosine squared theta minus sine squared theta. Equivalently, one minus two sine squared theta, or two cosine squared theta minus one. For tangent, substitute tangent of two theta equals two tangent theta over one minus tangent squared theta, where defined. These are the double-angle formulas: one repeated-angle idea. So how does one angle produce all three? Write two theta as theta plus theta, then apply the matching addition formula.