Binomial expansion / trigonometry question

I am completely stuck on the attached question (apparently it only requires knowledge of C4, but I have no idea of how to get the answer).

I've tried expressing sec(2θ) as 1/cos(2θ), and I've attempted to use the double angle formulae to express sin(θ/2) in terms of cosθ and sinθ.

If anyone can be at all helpful I would be really grateful.

Thank you

I am completely stuck on the attached question (apparently it only requires knowledge of C4, but I have no idea of how to get the answer).

I've tried expressing sec(2θ) as 1/cos(2θ), and I've attempted to use the double angle formulae to express sin(θ/2) in terms of cosθ and sinθ.

If anyone can be at all helpful I would be really grateful.

Thank you

$\sin\theta\approx \theta-\frac{1}{6}\theta^3$

Since this is true for all small theta, we can replace theta with anything, in particular we could replace it with theta/2 thus:

$\sin\theta/2\approx \theta/2-\frac{1}{6}(\theta/2)^3$

Similarly cos 2theta.

$\sin\theta\approx \theta-\frac{1}{6}\theta^3$

Since this is true for all small theta, we can replace theta with anything, in particular we could replace it with theta/2 thus:

$\sin\theta/2\approx \theta/2-\frac{1}{6}(\theta/2)^3$

Similarly cos 2theta.

Is it also allowable to use

$\sin^2 \dfrac{1}{2} \theta = \dfrac{1 - \cos \theta}{2}$