Add or subtract as indicated; then use fundamental identities to simplify the expression below and determine which of the following is not equivalent. $\sin x + 1 + \frac { 1 } { \sin x - 1 }$
A) $\frac { \sin ^ { 2 } x } { \sin x - 1 }$
B) $\frac { 1 - \cos ^ { 2 } x } { \sin x - 1 }$
C) $\frac { 1 + \cos ^ { 2 } x } { \sin x - 1 }$
D) $\frac { \sec x - \cos x } { \tan x - \sec x }$
E) $\frac { \csc x - \cot x \cos x } { 1 - \csc x }$

Question

Quizplus · Accepted Answer

The expression simplifies to $\frac{\sin^2x}{\sin x - 1}$ using the identity $\sin^2x + \cos^2x = 1$. Choices A, B, D, and E can be shown to be equivalent to this expression through various trigonometric identities and simplifications. However, choice C, $\frac{1 + \cos^2x}{\sin x - 1}$, does not simplify to the same form due to the presence of $1 + \cos^2x$ in the numerator, which does not directly relate to $\sin^2x$ without subtracting $\cos^2x$ from 1, making it not equivalent to the given expression.

Add or Subtract as Indicated; Then Use Fundamental Identities to Simplify