Prove the identities for double angles in trigonometry.

The double angle identities in trigonometry can be proven using basic algebraic manipulation and the Pythagorean identities.

The first set of double angle identities are:

sin(2θ) = 2sin(θ)cos(θ)
cos(2θ) = cos²(θ) - sin²(θ)

To prove the first identity, we start with the product-to-sum identity:

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

Setting A = B = θ, we get:

sin(2θ) = sin(θ + θ) = sin(θ)cos(θ) + cos(θ)sin(θ) = 2sin(θ)cos(θ)

To prove the second identity, we use the Pythagorean identity:

sin²(θ) + cos²(θ) = 1

Multiplying both sides by cos²(θ), we get:

cos²(θ)sin²(θ) + cos⁴(θ) = cos²(θ)

Rearranging and simplifying, we get:

cos²(θ)(1 - sin²(θ)) = cos²(θ) - cos²(θ)sin²(θ)

Using the Pythagorean identity again, we can substitute 1 - sin²(θ) for cos²(θ):

cos²(θ)cos²(θ) = cos²(θ) - cos²(θ)sin²(θ)

Dividing both sides by cos²(θ), we get:

cos²(θ) - sin²(θ) = cos(2θ)

Therefore, cos(2θ) = cos²(θ) - sin²(θ).

The second set of double angle identities are:

tan(2θ) = (2tan(θ))/(1 - tan²(θ))
cot(2θ) = (cot²(θ) - 1)/(2cot(θ))

To prove the first identity, we use the identity:

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

Setting A = B = θ, we get:

tan(2θ) = (tan(θ) + tan(θ))/(1 - tan²(θ)) = (2tan(θ))/(1 - tan²(θ))

To prove the second identity, we use the identity:

cot