Can You Solve This Trigonometric Identity?

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The discussion focuses on proving the trigonometric identity (cos A = cos B)^2 + (sin A + sin B)^2 = 2[1 + cos(A-B)]. A user expresses confusion about handling squared terms and seeks assistance in simplifying both sides of the equation. Other participants provide guidance on expanding the squared terms correctly and applying fundamental trigonometric identities, such as cos^2A + sin^2A = 1. They emphasize the importance of correctly expanding (cos A + cos B)^2 and (sin A + sin B)^2 to facilitate the proof. Ultimately, the user feels more confident after receiving hints and support from the community.
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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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First, do you know what (a+b)^2=...?? you have not expanded well. Give it another shot.

(cosA+cosB)^2=...?...

(sinA+sinB)^2=...?

After you expand this, remember also that:

cos^2A+sin^2A=1, and, sin^2B+cos^2B=1

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

2+ 2cosBcosA+2sinAsinB now the answer should be obvious, right?
 
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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...?
 
(sinA+sinB)^2=sin^2A+2sinAsinB+sin^2B...?

also

(cosA+cosB)^2=cos^2A+2cosAcosB+cos^2B


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

cos(A-B)=cosAcosB+sinAsinB
 
I think I can I didnt know that before, I will try to work it out now!
 
stuck said:
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.
 
I got it, Thank you!
 
stuck said:
I got it, Thank you!

You're welcome. THis is what PF is for!
 
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