Take a look at the identities below.

\[ \cos^2 \frac{\theta}{2} \equiv \frac{1}{2}(1+\cos \theta) \quad \quad \quad \sin^2 \frac{\theta}{2} \equiv \frac{1}{2}(1-\cos \theta)\]

You may well know enough trigonometric identities to be able to prove these results algebraically, but you could also prove them using geometry. We have provided some diagrams that may help you to prove the result for \(\cos^2 \frac{\theta}{2}\). Can you link the diagrams together to form a proof?

You may find it helpful to group the diagrams together in different ways or look for links between pairs of diagrams. You don’t need to use all the diagrams in your proof and you may prefer to add some of your own diagrams. The diagrams are available as a print out. There is an extra card in case you’d like to include another diagram in your proof.

On a circle of radius 1 centre O, C is the point where the horizontal diameter meets the circle on the left, A is a point on the upper half of the circle, and theta is the angle between the line AO and the horizontal diameter of the circle.
A and C are the points as in the diagram on the left and B is the perpendicular foot of A on the horizontal diameter of the circle, so that that ABC is a right angled triangle with right angle at B.
Circle showing the right angled triangle ABO, the angle theta at O, and the hypotenuse of length 1.
Circle showing the triangle AOC, and the point D which is the perpendicular foot of O on the line AC, dividing the triangle AOC into two right angled triangles.
Similar to the top right diagram but with the point C marked and the line CB added.

It might be easier to prove \(1+\cos \theta \equiv 2\cos^2 \frac{\theta}{2}\).

Can you relate the diagrams to either the left-hand side of the identity or to the right-hand side?

Is there a diagram that shows a length that corresponds to \(1+\cos \theta\)?

Amalgamation of all the above diagrams

Can you prove the result for \(\sin^2 \frac{\theta}{2}\) in a similar way?