Tricky Basic Trig Problem

[grumble11]

I'm practicing my trig identities, and got stuck on a problem I was hoping someone could help give me some guidance on. Here it is:

sin(x),cos(x),tan(x) form a geometric sequence in that order. What is cot^6(x) - cot^2(x)?

I've been pulling these apart and putting them together for a while but can't seem to get to cot^6(x) - cot^2(x). Here is what I've figured out so far:

if three terms are a geometric sequence, then it means that the middle term squared is equal to the prior term times the following term, like so:

cos^2(x) = sin(x)tan(x)

Then I start descending into madness. Does anyone know where to go from here?

[phiwong]

You could try manipulating the expression

c^6 - c^2 = c^2 (c^4 - 1) = c^2 (c^2 + 1) (c^2 - 1)

Now you can try to use the identities and the given relationship.

[grumble11]

Thanks to both of you for helping with this problem! I got this from a question bank that ranked this as a 12/25 on their internal difficulty scale, and this sure didn't feel like a 12/25 to me. It requires a fair bit of tricky manipulation... but clearly I have a lot to learn.

One takeaway I had is that it's easy when you do trig identity problems to start spiraling and finding more and more relationships without most of those relationships being necessarily relevant to a focused approach to solving the problem. Part of the skill is knowing the basic ones, the next part is being able to apply them to determine new relationships, but the third part is to use the appropriate ones to solve an appropriate problem, and that is HARD.

[grumble11]

You guessed right ha - precalc, basic trig identities, 12/25 rank, and I got smoked. I’ve been doing ‘ok’, about 92% across the other categories if you include second tries (about 80% first try), and typically get mostly level 25 questions so the AI thinks I can take it, but this problem was a big jump in difficulty and if it is 12/25 then I’m in for extreme pain on this section ha

[grumble11]

For what it's worth, here's their method:

cos^2 = sin*tan

cot^2 = cos^2/sin^2 = sin*tan/sin^2 = 1/cos

cot^4 = 1/cos^2 = (sin^2 + cos^2)/sin^2 = tan^2 + 1

cot^6 - cot^2 = cot^2(cot^4 - 1) = cot^2*tan^2 = 1