What are all the proofs of cos(a) sin(a + b) = sin(b) + cos(a + b) sin(a) that don't use trigonometric identities?
#Proof of cos(𝘢) sin(𝘢 + 𝘣) = sin(𝘣) + cos(𝘢 + 𝘣) sin(𝘢)
rancid lantern
torn iglooBOT
What are all the proofs of cos(*a*) sin(*a* + *b*) = sin(*b*) + cos(*a* + *b*) s...
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paper warren
Are there any other proofs?
since theyre using chatgpt they dont have a brain and cant actually think
does this whole thing expand to be equivalent to just cos(a)(the expansion of sin(a+b))? there are geometric proofs for those
paper warren
stop using chat gpt. if you know the content help if not get out
rancid lantern
since theyre using chatgpt they dont have a brain and cant actually think
does t...
I don't think so, but are there any geometric proofs for cos(a) sin(a + b) = sin(b) + cos(a + b) sin(a)?
But I'd also be okay with a geometric proof of cos(a) sin(a − b) = −sin(b) + cos(a − b) sin(a).
Perhaps something involving two triangles?
rancid lantern
@paper warren, is this what you mean?
cos(a) sin(a + b) = sin(b) + cos(a + b) sin(a)
cos(a) (sin(a) cos(b) + cos(a) sin(b)) = sin(b) + cos(a + b) sin(a)
cos(a) sin(a) cos(b) + cos²(a) sin(b) = sin(b) + cos(a + b) sin(a)
cos(a) sin(a) cos(b) + cos²(a) sin(b) = sin(b) + (cos(a) cos(b) – sin(a) sin(b)) sin(a)
cos(a) sin(a) cos(b) + cos²(a) sin(b) = sin(b) + sin(a) cos(a) cos(b) – sin²(a) sin(b)
cos²(a) sin(b) = sin(b) – sin²(a) sin(b)
cos²(a) sin(b) + sin²(a) sin(b) = sin(b)
sin(b) (cos²(a) + sin²(a)) = sin(b)
cos²(a) + sin²(a) = 1, sin(b) ≠ 0
rancid lantern
Where did you find this?
runic saddle
Where did you find this?
Wikipedia
mellow light
Let f:a ↦ LHS-RHS. We have f(0)=0 and f’(a)=0
minor viper
Let f:a ↦ LHS-RHS. We have f(0)=0 and f’(a)=0
That's assuming they know calculus
mellow light
They want all the proofs, it is one of them. Another proof maybe outside their knowledge would use complex numbers. If trigonometric identities were allowed, it would be shorter to expand sin(b+a-a). And then my proof would allow to prove the trigonometric identities like sin(a+b), but only to check them, not to find them in a constructive manner, except for finding one from another
YAP with power series