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We received several different solutions, all of which were correct and elegant.
Joshua from Bohunt Sixth Form in the UK used the cosine rule and Pythagoras's Theorem. Click to see Joshua's work.
Soumya from King's Maths School used the cosine rule and the sine rule. Click to see Soumya's work.
Dylan from Brooke Weston in the UK derived the formulas using the unlabelled angle from the original diagram. Click to see Dylan's work.
Nishad from Thomas Estley Community College in England split the triangles into lots of smaller right-angled triangles to prove the identities. Click to see Nishad's work.