Is there an easier/another way to prove the trig identity Cos(a+b)= cos(a)cos(b)-sin(a)sin(b).I don't want to teach students to memorize this,and the traditional proof uses the unit circle, the distance formula, and the law of cosines. Too much information for one lesson.
Of course you dont want to teach students to memorize the identity! You should be more than happy to to use the unit circle, distance formulas etc. as long as you do this as a mathematical journey say starting with Pythagorous ... .
The point of learning mathematics involves more than getting and using identities. They will be gone from the students memories soon enough. It is the ideas, the logic and the problem solving strategies that are important and which remain. To say "Too much information for one lesson" is disturbing. Since when do we teach ideas in one hour packages? If it takes more to do it properly, it takes more to do it properly.
Maybe you could start from looking at why you want the identity and then work backwards piece by piece until you found a justification for it.
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