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Question
I could use some help with this question. I am prepping for my ACTs, but have really struggled with this question. I think just having the whole process of how to get the right answer would help.

I am given this expression: cot^(2)a - cot^(2)a * cos^(2)a

I am told to first factor, then use the fundamental identities to determine which of the following values our expression is NOT equivalent to.
sin^2((pi/2)-a)
tan^(2)a
cos^(2)a
1/sec^(2)a
1-sin^(2)a

Answer
Hi Donny,
cotē(a) - cotē(a).cosē(a) = cotē(a) [1 - cosē(a)]
= cotē(a) . sinē(a)        (since cosē(a) + sinē(a) = 1)
= sinē(a) / tanē(a)       (since tan(a) = 1/cot(a))
= cosē(a)          (since tan(a) = sin(a)/cos(a))

All we need to do now is compare it with the provided expressions. Of course by inspection we can already see that it isn't equivalent to the second option, tanē(a).
Anyway,
1) sin(π/2 - a) = cos(a)       (an identity)
so,
sinē(π/2 - a) = cosē(a)

2) Clearly, tanē(a) ≠ cosē(a)

3) Without any doubt, cosē(a) = cosē(a)

4) sec(a) = 1/cos(a)
so,
1/secē(a) = cosē(a)

5) cosē(a) + sinē(a) = 1       (an identity)
therefore,
1 - sinē(a) = cosē(a)

And there you have it.

Regards

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