I need some help-Trig Identities

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  • #1
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Prove the following identities:

(cos A = cos B)^2 + (sin A + sin B)^2 = 2[1+cos(A-B)]



I'm really a mess at this stuff. I missed a few important days and fell behind, so I don't reeeally know what to do when things start getting squared and whatnot., but I tried! :bugeye:

Left Side
(cos A + Cos B)^2 + (sin A + sin B)^2
(cos(^2)A+cos(^2)B = Sin(^2)A+sin(^2)B
(cos (^2)A+cos(^2)B+ (1+-cos(^2)A + (1+-cos(^2)B)
2


Right Side
2+2cos(A-B)
2(cosAcosB+sinAsinB) +2

But that's as far as I can get. I can't find a way to make both sides equal, and I'm not even sure if my left side is correct...

Please help me if you can! :shy:
 
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Answers and Replies

  • #2
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First, do you know what [tex](a+b)^2=...???????[/tex] you have not expanded well. Give it another shot.

[tex] (cosA+cosB)^2=...?....[/tex]

[tex](sinA+sinB)^2=....?[/tex]

After you expand this, remember also that:

[tex] cos^2A+sin^2A=1, and, sin^2B+cos^2B=1[/tex]

And somewhere in between you will end up with sth like this

[tex]2+ 2cosBcosA+2sinAsinB[/tex] now the answer should be obvious, right?
 
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  • #3
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Urk, I thought that because in my third step on the left I had (cos^2A) and (-cos ^2A), that they would cancel...?
 
  • #4
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[tex](sinA+sinB)^2=sin^2A+2sinAsinB+sin^2B....?[/tex]

also

[tex] (cosA+cosB)^2=cos^2A+2cosAcosB+cos^2B[/tex]


Can you go from here now?

Just use the hints that i gave you on my post #2

also, another hint, altough you should have figured this out by yourself

[tex]cos(A-B)=cosAcosB+sinAsinB[/tex]
 
  • #5
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I think I can I didnt know that before, I will try to work it out now!
 
  • #6
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I think I can I didnt know that before, I will try to work it out now!
I wrote in there everything you need to do that problem, you only need to put all the hints together.
 
  • #7
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I got it, Thank you!
 
  • #8
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I got it, Thank you!
You're welcome. THis is what PF is for!
 

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