Prove the identities for half-angles in trigonometry.

To prove the identities for half-angles in trigonometry, we can use the double-angle formulae and some algebraic manipulation.

Firstly, we can use the double-angle formula for cosine to obtain:

cos(2θ) = 2cos²(θ) - 1

Rearranging this formula, we get:

cos²(θ) = (cos(2θ) + 1) / 2

Now, we can substitute θ/2 for θ in the above formula to obtain:

cos²(θ/2) = (cos(θ) + 1) / 2

This is the identity for half-angle of cosine.

Similarly, we can use the double-angle formula for sine to obtain:

sin(2θ) = 2sin(θ)cos(θ)

Rearranging this formula, we get:

sin(θ) = 2sin(θ/2)cos(θ/2)

Dividing both sides by cos(θ/2), we get:

tan(θ/2) = sin(θ/2) / cos(θ/2) = (1 - cos(θ)) / sin(θ)

This is the identity for half-angle of tangent.

Finally, we can use the identity for half-angle of sine, which is obtained by rearranging the above formula for tan(θ/2) and using the Pythagorean identity:

sin²(θ/2) = (1 - cos(θ)) / 2

In summary, the identities for half-angles in trigonometry are:

cos²(θ/2) = (cos(θ) + 1) / 2
tan(θ/2) = (1 - cos(θ)) / sin(θ)
sin²(θ/2) = (1 - cos(θ)) / 2