Verify that the trigonometric equation is an identity cos x

Verify that the trigonometric equation is an identity. cos x - 1/cos x + 1 - cos x + 1/cos x - 1 = 4 cot x csc x Which of the following statements establishes the identity? A. cos x - 1/cos x + 1 - cos x + 1/cos x - 1 = (1 - sec x)^2 - (1 + sec x)^2/(1 + sec x)(1 - sec x) = -4 sec x/1- sec^2 x = 4 sec x/tan^2 x = 4 cot x csc x B. cos x - 1/cos + 1 - cos x 1/cos x - 1 = (cos x - 1)^2 - (cos x + 1)^2/(cos x + 1) (cos x -1) = -4 cs x/cos^2 x - 1 = 4 cos x/sin^2 x = 4 cot x csc x C. cos x - 1/cos x + 1 - cos x + 1/cos x - 1 = (sin x - 1)^2 - (sin x + 1)^2/(sin x + 1)(sin x -1) = -4 sin x/sin^2 x - 1 = 4 sin x/cos^2 = 4 cot x csc x D. cos x - 1/cos x + 1 - cos x + 1/cos x - 1= (1 - csc x)^2/(1 + csc x)(1 - csc) = -4 csc x/1 - csc^2 x = 4 csc x/cot^2 x = 4 cot x csc x

Solution

LHS = [(cos x - 1)/(cos x + 1)] - [(cos x + 1)/(cos x - 1)]

= [(cos x - 1)^2 - (cos x + 1)^2]/(cos^2 x - 1)

= [cos^2 x + 1 - 2*cos x - cos^2 x - 1 - 2*cos x]/(-2*sin^2 x)

= -4*cos x/(-2*sin^2 x)

= 2*cos x/sin^2 x

= 2*cot x*csc x = RHS