I'm having a hard time remembering where the cos(theta)^4 is coming from.
I know one is because the light hits the film at an angle; the only way to capture the full radiance is to be perpendicular to the light.
Another cos(theta) is because the cross-sectional area of the lens gets smaller with theta; if we had theta-89 degrees then our lens would just be a little sliver.
Where do the other to cos(theta) terms come from?
Please see the notes on the measurement equation from a previous lecture. First ask yourself, why is there a cos^2(theta) (answer: change of variables from an integral over solid angle to an integral over area of the aperture). Then the other two cos^2(theta)'s come from rewriting the 1/R^2 term in terms of d.
@ak-47 The other two cos(theta) comes from r = d / cos(theta). And we are dividing it by r^2 to get dW from dA, so its' cos(theta)^4.
I hope it is clear.