Prove the double angle formulae in trigonometry.

The double angle formulae in trigonometry can be proven using the sum and difference formulae.

To prove the double angle formulae, we start with the sum formulae for sine and cosine:

sin(A + B) = sin(A)cos(B) + cos(A)sin(B)
cos(A + B) = cos(A)cos(B) - sin(A)sin(B)

We can use these formulae to derive the double angle formulae for sine and cosine:

sin(2A) = sin(A + A) = sin(A)cos(A) + cos(A)sin(A) = 2sin(A)cos(A)
cos(2A) = cos(A + A) = cos(A)cos(A) - sin(A)sin(A) = cos^2(A) - sin^2(A)

To prove the double angle formula for tangent, we start with the formula:

tan(A + B) = (tan(A) + tan(B))/(1 - tan(A)tan(B))

We can substitute A = B in this formula to get:

tan(2A) = (tan(A) + tan(A))/(1 - tan(A)tan(A)) = 2tan(A)/(1 - tan^2(A))

Therefore, the double angle formulae for sine, cosine, and tangent are:

sin(2A) = 2sin(A)cos(A)
cos(2A) = cos^2(A) - sin^2(A)
tan(2A) = 2tan(A)/(1 - tan^2(A))

These formulae are useful in simplifying trigonometric expressions and solving trigonometric equations.