Factor; then use fundamental identities to simplify the expression below and determine which of the following is not equivalent.
$\cot ^ { 2 } \alpha + \cot ^ { 2 } \alpha \cdot \tan ^ { 2 } \alpha$
A) $\csc ^ { 2 } \alpha$
B) $1 + \cot ^ { 2 } \alpha$
C) $\sec ^ { 2 } \alpha$
D) $1 / \cos ^ { 2 } \left( \frac { \pi } { 2 } - \alpha \right)$
E) $\frac { 1 } { 1 - \cos ^ { 2 } \alpha }$

Question

Factor; then use fundamental identities to simplify the expression below and determine which of the following is not equivalent.  
  $\cot ^ { 2 } \alpha + \cot ^ { 2 } \alpha \cdot \tan ^ { 2 } \alpha$
A)  $\csc ^ { 2 } \alpha$
B)  $1 + \cot ^ { 2 } \alpha$
C)  $\sec ^ { 2 } \alpha$
D)  $1 / \cos ^ { 2 } \left( \frac { \pi } { 2 } - \alpha \right)$
E)  $\frac { 1 } { 1 - \cos ^ { 2 } \alpha }$

Quizplus · Accepted Answer

The given expression simplifies to $\csc^2 \alpha$ using trigonometric identities. $\cot^2 \alpha + \cot^2 \alpha \cdot \tan^2 \alpha = \cot^2 \alpha (1 + \tan^2 \alpha) = \cot^2 \alpha \cdot \sec^2 \alpha$. However, $\cot^2 \alpha \cdot \sec^2 \alpha$ does not simplify directly to $\sec^2 \alpha$, making choice C incorrect. Choices A, B, D, and E can be shown to be equivalent to $\csc^2 \alpha$ through various trigonometric identities and transformations.

Factor; Then Use Fundamental Identities to Simplify the Expression Below cot⁡2α+cot⁡2α⋅tan⁡2α\cot ^ { 2 } \alpha + \cot ^ { 2 } \alpha \cdot \tan ^ { 2 } \alphacot2α+cot2α⋅tan2α

Factor; Then Use Fundamental Identities to Simplify the Expression Below $\cot ^ { 2 } \alpha + \cot ^ { 2 } \alpha \cdot \tan ^ { 2 } \alpha$