Prove that the following is true for all angles θ:

Question

a)(1 − sin θ) (1 + sin θ) = cos^2θ

b) tan θ + cot θ = tan θ · csc^2θ?

Answer 1

a) Left hand side identity = [1 - sin(θ)] [1 + sin(θ)]

{Apply the formula ( a+ b) (a - b) = a²  - b² , In this case a = 1 , b = sin(θ)}

                                 = [ (1)²  - sin²(θ)]

                                 = 1 - sin²(θ)

{ From the pythgorean identities sin²(x) + cos²(x) = 1}

                                = cos²(θ)

                                = Right hand side identity.

b) Left hand side identity = tan(θ) + cot(θ)

{ Reciprocal identity of tan(θ) =1/cot(θ) }

                                   = [1/ cot(θ)] + cot(θ)

                                   = [ 1+ cot²(θ) ]/cot(θ)

{ From the pythgorean identities 1+ cot²(x) = csc²(x) }

                                   = csc²(θ)/ cot(θ)

                                   = [1/ cot(θ)] [csc²(θ)]

                                   = tan(θ) csc²(θ)

                                   = Right hand side identity.