Particularly General
Why do this problem?
This problem gives practice and insight into manipulating and constructing algebraic identities. It is based upon the idea of 'proof by generic example': the proof method of a particular example encapsulates in a clear way all of the key points of a proof of a more general case. Manipulation of algebraic identities is always good fun and will give very solid general preparation for the more challenging end of school mathematics examinations, such as Further Mathematics and STEP. It can also be used to help students to understand the differences in the uses of direct proof and proof by induction.
Possible approach
Key questions
What is the difference between an identity and an equation?
What method of proof can you use for the specific examples given?
How can you expand the brackets or reduce the terms in a step by step way? Can you see how this procedure might generalise?
Possible extension
Using the ideas gained from solving this problem, can students construct some new identities of their own?
Possible support
Suggest the use of difference of two squares and a trigonometric double angle formula.
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