Is it possible to prove the cosine addition formula with the Cosine Law and distance formula?

[Four Muffins]

Hi, I got stuck on question 86 for hours, and I'm not sure if it is because I misinterpreted it. I was able to solve 85, and thought that 86 was referring to the Cosine Law and distance formula. I tried to solve it in similar ways to 85, but was unable to. I looked for other proofs of the cosine addition formula for a hint, and found nothing similar. Was what I was trying to do impossible? Was the 'formula in Exercise 85' likely referring to the difference formula, not the Cosine Law and distance formula?



Question 85 proof

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One of many question 86 attempts. I tried as many variations on the 85 method as I could think of.

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Complete proof (maybe?)

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[Dr.Peterson]

Was the 'formula in Exercise 85' likely referring to the difference formula, not the Cosine Law and distance formula?

Of course it is! It says, "use the formula", meaning the formula shown and proved there, not "use a similar method".

Given the subtraction formula, it is easy to derive the addition formula, as you did at the end.

In your attempt at 86, your picture is labeled incorrectly. A is [imath](\cos(\alpha+\beta),\sin(\alpha+\beta))[/imath]. Other things are wrong after that. But if you are more careful, a similar method can be used. (One option is to put [imath]\alpha[/imath] below the axis, rather than above [imath]\beta[/imath].)

 

[Four Muffins]

As long as I was being silly, and not just being bad at it, that's okay. Thanks I'll give the negative alpha method a go too.