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trignometry question (inverse functions) watch

1. Prove that

so
let

let

therefore

now to find tan(4x) is immensely long does anyone know of a quicker way to do this?

Thanks
2. Double-angle forumla twice can't be that much effort?
3. Forgive me if I'm being stupid but surely tan(4x) = tan (pi -y) = -tan y

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So this means tan(4x+y)=0 => 4x+y = npi

and y<0.5pi and 4x<pi
4. Alternatively this could be done with complex numbers; after rewriting arccot 2, we wish to prove that:

Noting that (taking a principal argument), what if we consider the argument of the expression in two different ways? This is in fact almost a Machin-Like Formula.
5. (Original post by rbnphlp)
...
(Original post by electriic_ink)
Forgive me if I'm being stupid but surely tan(4x) = tan (pi -y) = -tan y
Perhaps I am misreading this, but it seems to me that this result is assuming the very thing that you need to prove.
6. (Original post by ghostwalker)
Perhaps I am misreading this, but it seems to me that this result is assuming the very thing that you need to prove.
No, you're not. I've just realised the OP wrote 4x + y = pi. I misread that as fact but actually that is what he's supposed to be proving.
7. (Original post by ghostwalker)
Perhaps I am misreading this, but it seems to me that this result is assuming the very thing that you need to prove.

@ unbounded:thanks unbounded seems like you had the alternative solution after all
8. (Original post by rbnphlp)

@ unbounded:thanks unbounded seems like you had the alternative solution after all
Actually it really isn't that much work to use the double angle formula twice on tan(x)=1/2. You can almost work out that tan(2x)=4/3 in your head.

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