Determine the intensity of light passing through a polarizer-analyzer combination with an angle of 45° between them.

Answer 1

assume the incoming light is coming an at an angle theta to the first filter, and assume it is not circular polarization.
light getting through
I*costheta
now the light getting thru the second filter is
I*cosTheta*cos(theta+45)

now if I remember my trig identities that equals
I*(cosTheta(cos(theta)cos45-sinThetasin45)=
I(cos^2theta*cos45-cosTheta*sinTheta*sin45)=
I(.707)(cos^2Theta -cosTheta*SinTheta)=

Now, comparing that to the original incident light, it depends on the incident angle with the first plate. If Theta is zero, then .707 gets thru. If Theta is 90 deg, zero gets thru. You can see the graph here.
http://www.wolframalpha.com/input/?i=plot+y-%3D.707((cos(x)*cos(x)-sin(x)*cos(x))