Polarization Filters the Direction of Vibration
The electric field of light shakes at right angles to the way it travels. A polarizer is a sieve that keeps only one direction of that shaking. Cross two of them and the light vanishes, yet slide a third between them and it springs back to life. Malus's law explains all of it with a single angle.
Light is a transverse wave: its electric field oscillates at right angles to the direction of travel. A polarizer passes only the part of that oscillation along its transmission axis. The amplitude drops to A0 cos θ, and because intensity is amplitude squared, I = I0 cos2 θ. This is Malus's law. Polarized sunglasses use this very principle, using a vertical transmission axis to block the glare that reflects horizontally off water or a road.
Put a second polarizer in the beam and rotate the rear one, the analyzer. When the two axes are parallel (0 degrees) almost everything passes; when they are crossed (90 degrees) it goes completely dark. The transmitted intensity traces the cos2 θ curve. A liquid crystal display is built on this pair of polarizers, with the liquid crystal between them twisting the direction of vibration to set the brightness of each pixel.
Two polarizers crossed at right angles block the light completely. Yet slip a third one at an angle between them and the light returns. Each filter projects the field onto a new axis, so the middle one hands the last filter a component it can pass. The output peaks when the middle sits at 45 degrees, letting one eighth of the incoming light through. If a polarizer only subtracted light, adding one could only darken the beam, so the fact that it brightens instead reveals that a polarizer realigns the direction of vibration onto a new axis.
Natural light oscillates evenly in every direction. Passing the first polarizer cuts its intensity to exactly one half no matter which way the axis points, and afterward the light is linearly polarized along one direction. From the next filter on, it obeys Malus's law, cos2 θ. The reason natural light drops to exactly half is that every angle is mixed in evenly, and the average of cos2 θ over a full turn is precisely one half.
The root of Malus's law is a simple projection. Drop the electric field vector onto the transmission axis like a shadow and its length is cos θ of the original. The intensity is that shadow squared, so it is cos2 θ. When the vector tilts to 45 degrees the shadow shrinks to about 0.707 of its length, and squaring that gives exactly half the intensity. Rotate the vector and watch the shadow shrink and grow.