Duplication and linearisation formulas

Doubling Formulas

By setting b = a in the addition formulas, we obtain the double-angle formulas:

sin(2a) = 2sin(a)cos(a) cos(2a) = cos²(a) - sin²(a) = 2cos²(a) - 1 = 1 - 2sin²(a)

These three expressions for cos(2a) are equivalent; we choose the one that simplifies the calculation based on the available data (for example, if we only know cos(a), we use 2*cos²(a) - 1).

Linearisation

Using the preceding formulas, we isolate cos²(a) and sin²(a), which allows us to linearise an expression (remove the squares):

cos²(a) = (1 + cos(2a))/2 sin²(a) = (1 - cos(2a))/2

Example: to simplify an expression containing sin²(x), we replace it with (1 - cos(2x))/2 in order to obtain a form without squares, which is easier to analyse or integrate.

Common pitfall

Do not confuse sin(2a) with 2sin(a): these are two different expressions. For example, for a = π/6, sin(2a) = sin(π/3) = √3/2, whereas 2sin(a) = 2*(1/2) = 1. The two values are clearly distinct, so always check which argument the factor 2 applies to.