Maths - Any Ideas

Question

Hello,

There are some questions that I am not sure how to tackle:

1) sin3A = 3sinA - 4sin^3A

2) cos3A = 4cos^3A - 3cosA

3) cotθ = (sin2θ) / (1 - cos2θ)

4) cos^4θ - sin^4θ = cos2θ

Thank you in advance.

Answer 1

(1). sin3A = 3sinA - 4sin^3A.

Left hand side identity = sin 3A

                                   = sin(2A + A)

Apply Sum and Difference Formula : sin(u+ v) = sin(u) cos(v) + cos(u) sin(v).

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

Apply Double angle Formulas : sin(2u) = 2 sin(u) cos(u) and cos(2u) = 1 - 2sin²(u).

                                   = [2 sin(A) cos(A)] cos(A) + [1 - 2sin²(A)] sin(A)

                                   = 2 sin(A) cos²(A) + sin(A) - 2sin³(A)

                                   = 2 sin(A) [1 - sin²(A)] + sin(A) - 2sin³(A)

                                   = 2 sin(A) - 2sin³(A) + sin(A) - 2sin³(A)

                                   = 3sin(A) - 2sin³(A) - 2sin³(A)

                                   = 3sin(A) - 4sin³(A).

                                   = Right hand side identity.

Answer 2

(2). cos3A = 4cos^3A - 3cosA.

Left hand side identity = cos 3A

                                   = cos(2A + A)

Apply Sum and Difference Formula : cos(u+ v) = cos(u) cos(v) - sin(u) sin(v).

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

Apply Double angle Formulas : sin(2u) = 2 sin(u) cos(u) and cos(2u) = 2 cos²(u) - 1.

                                   = [2 cos²(A) - 1] cos(A) - 2 sin(A) cosA sin(A)

                                   = 2 cos³(A) - cos(A) - 2 sin²(A) cosA

                                   = 2 cos³(A) - cos(A) - 2[1- cos²(A)] cosA

                                   = 2 cos³(A) - cos(A) - 2 cosA + 2 cos³(A)

                                   = 4 cos³(A) - 3 cosA

                                   = Right hand side identity.

Answer 3

(3). cotθ = (sin2θ) / (1 - cos2θ)

Left hand side identity = cot(θ)

                                   = cos(θ)/sin(θ)

                                   = [ cos(θ)/sin(θ) ] * [ 2sin(θ) / 2sin(θ) ]

                                   = 2 cos(θ)sin(θ) / 2sin²(θ)

Apply Double angle Formulas : 2 sin(u) cos(u) = sin(2u) and 2sin²(u) = 1 - cos(2u).

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

                                   = Right hand side identity.

Answer 4

(4). cos^4θ - sin^4θ = cos2θ

Left hand side identity = cos⁴(θ) - sin⁴(θ)

Apply Difference of Two Squares : a²- b² = (a + b)(a - b).

                                   = [cos²(θ) + sin²(θ)] * [cos²(θ) - sin²(θ)]

Apply Pythagorean identity : sin²(u) + cos²(u) = 1.

                                   = [cos²(θ) - sin²(θ)]

Apply Double angle Formulas : cos(2u) = cos²(u) - sin²(u).

                                   = cos(2θ).

                                   = Right hand side identity.