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Proving Half-angle Formulae

Age 16 to 18 Challenge Level:

This resource is from Underground Mathematics.

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.



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

This is an Underground Mathematics resource.

Underground Mathematics is funded by a grant from the UK Department for Education and provides free web-based resources that support the teaching and learning of post-16 mathematics. It started in 2012 as the Cambridge Mathematics Education Project (CMEP).

Visit the site at to find more resources, which also offer suggestions, solutions and teacher notes to help with their use in the classroom.