We will extend our knowledge of compound angle formulas to include the double angle formulas.

These formulas are special cases of the angle sum formulas studied in the previous module.

Consider that \(\sin(2\theta)\) can be expressed as \(\sin(\theta + \theta)\).

Using the angle sum formula for sine, \(\sin(A + B) = \sin(A)\cos(B) + \cos(A)\sin(B)\), we have

Thus, the double angle formula for sine is \(\sin(2\theta) = 2\sin(\theta)\cos(\theta)\).

Similarly, we can develop a formula for \(\cos(2\theta)\) using the angle sum formula for cosine, \(\cos(A + B) = \cos(A)\cos(B) - \sin(A)\sin(B)\).

We have,

Making appropriate substitutions using the Pythagorean identity, \(\sin^2(\theta) + \cos^2(\theta) = 1\), the double angle formula for cosine can be expressed in two other ways.

Using \({\color{BrickRed}\cos^2(\theta) = 1 - \sin^2(\theta)}\),

Using \({\color{BrickRed}\sin^2(\theta) = 1 - \cos^2(\theta)}\),

Thus, the double angle formula for cosine is given by

Using the angle sum formula for tangent,

express \(\tan(2\theta)\) in terms of \(\tan(\theta)\).

Using the angle sum formula for tangent, we have

Alternatively, using the quotient identity, we have