Isaac Physics: Through the looking glass

https://isaacphysics.org/questions/through_the_looking_glass?stage=a_level

I have solved part A and B but don't know about C.
I thought you could use the sine rule on the triangle formed by h, R and the refracted ray such that h/(180-theta)=R/(sin(theta-2*theta_g+90) and rearrange for h. However then the denominator is not in the required form of cos(2*theta_g-theta) and I don't think it's the correct way to get h anyway.
Any help would be appreciated.

Without looking at the question itself, starting from your expression, consider some simple trig. transformations.

E.g what's sin(x+90) ?

Then one other.

Without looking at the question itself, starting from your expression, consider some simple trig. transformations.

E.g what's sin(x+90) ?

Then one other.