Really hard Maths question Help plz

Hi i have been really struggling with this maths question:

https://isaacphysics.org/questions/maclaurin_series_trig1?board=606ebd4c-be70-47e4-9325-1221bbcf6584

can anyone figure it out ???
I am stuck on part c

What have you done for maclaurin expansion of cos?
When posting questions like this, it helps if you describe what you understand and what you're stuck on.

What have you done for maclaurin expansion of cos?
When posting questions like this, it helps if you describe what you understand and what you're stuck on.

yeah sorry should have been more specific. I have hardly even know how to start. i don't understand how the mgl and the 1- will effect the maclaurin expansion of cos, also I am not completely clear on the difference between maclaurin and a taylor expansion of cos.
thanks for your help so far

A Maclaurin series is a Taylor series about 0.
So for this question they're the same?
So what do you get for cos?

i got 1- (x^2)/(2 factorial)+(x^4)/(4factorial).......

When x is small, you just need the first two terms.
So subtract from 1 and multiply by mgl.
The answer is fairly simple.

ok thanks i think i am nearly there, I got -mgl/4 but that's not right what have I done wrong???

Not sure, you've not given your working.

o right sorry

1 - (1 - x^2/2) = ?
Where does the second 2! come from?

i wasnt sure how i could get sigma squared on its own so I split it into sigma squared over two factorial and -mgl over two factorial and then wrote sigma squared over to factoral as on over two factorial times sigma squared, but I think this might be where I went wrong

You've not answered the first hint.

Phi is just x here so you have something like
mglx^2 = (mgl)*(x^2)
You may be over thinking the second part.

but then wouldn't mgl be the answer and I have tried that already, how can I use that to find the expression

Err yes. I didn't give you the answer.
What did you do the first hint and what is the expression?

if mglx^2 = Ep = Ax^2 then wouldn't A = mgl?

Yes, so what do you actually get for Ep?

well then Ep would be mglx^2 and the A would be mglx^2/x^2 which would mean A = mgl but that's wrong so I have no idea what to do

Please post the working carefully for Ep.
Your previous attempt was close, apart from the two errors mentioned. I could give the answer, but tbh it's better for you to get your working sorted.