D. csc^2 x + sec^2 x = 1
The process for each option is to rewrite the equation, attempting to obtain the identity sin^2 x + cos^2 x = 1. In general convert each function to its equivalent using just sin and cos.
A. cos^2 x csc x - csc x = -sin x
cos^2 x * 1/sin x - 1/sin x = -sin x
(cos^2 x * 1/sin x - 1/sin x) * sin x = -sin x * sin x
cos^2 x * 1 - 1 = -sin^2 x
cos^2 x = -sin^2 x + 1
cos^2 x + sin^2 x = 1
Option A is an identity.
B. sin x(cot x + tan x) = sec x
sin x(cos x/sin x + sin x/cos x) = 1/cos x
cos x + sin^2 x/cos x = 1/cos x
cos^2 x + sin^2 x = 1
Option B is an identity.
C. cos^2 x - sin^2 x = 1- 2sin^2 x
cos^2 x - sin^2 x + 2sin^2 x = 1- 2sin^2 x + 2sin^2 x
cos^2 x + sin^2 x = 1
Option C is an identity.
D. csc^2 x + sec^2 x = 1
1/sin^2 x + 1/cos^2 x = 1
cos^2 x/(cos ^2 x sin^2 x) + sin^2 x/(cos^2 x sin^2 x) = 1
(cos^2 x + sin^2 x)/(cos ^2 x sin^2 x) = 1
1/(cos ^2 x sin^2 x) = 1
1 = cos ^2 x sin^2 x
Option D is NOT an identity.
