Prove tan(4pi/11)+4sin(pi/11)=sqrt(11)

By Tkarthik on November 16, 1998 :

prove

tan(4p/11)+4sin(p/11)=

__
Ö11

By Alex Barnard on February 12, 1999 :

I don't know how much you know so if anything below doesn't make sense then please write back.

A very effective way to prove most trig identities like the one above is to use complex numbers. To do this we use the formula

e^{i x} = cos(x) + isin(x)

So cos(x) = (1/2)(e^{i x} + e^{-i x})

and sin(x) = (1/2i)(e^{i x} - e^{-i x})

So if I set x = ip/11, then the question above is to show that:

x⁴-x^-4
x⁴+x^-4
+2(x¹-x^-1)=i __
Ö11

Now if we multiply both sides by (x⁴ + x^-4) and then square each side and simplify then we get (after a little work which I'm certainly not typing out!):

4(x¹⁰ + x⁹ + x⁸ - x⁷ + x⁶ + 2x² + 1 + x^-10 + x^-9 + x^-8 - x^-7 + x^-6 + 2x^-2) = 0

Now x¹¹ = -1 (because x¹¹ = e^ip = -1)

So -1 = x¹1 = x⁹⁺² = x⁹ ×x² Þ x⁹ + x² = 0

And similarly -x⁷ = x^-4

So the long equation above becomes:

(x¹⁰ + x⁸ + x⁶ + x⁴ + x² + x^-10 + x^-8 + x^-6 + x^-4 + x^-2 + 1) = 0

So if we can prove this is true then the result you want will be true.

Now, x¹¹ = -1 means that x²² = 1

So 0 = x²² - 1 = (x-1)(1 + x + x² + x³ + ¼+ x²¹ )

= (x-1)(1+x)(1 + x² + x⁴ + x⁶ + ¼+ x²⁰ )

Now, clearly x isn't 1 or -1, so the above means that:

1 + x² + x⁴ + ¼+ x²⁰ = 0

Dividing by x¹⁰ (allowed as x isn't 0) gives:

x¹⁰ + x⁸ + x⁶ + x⁴ + x² + x^-10 + x^-8 + x^-6 + x^-4 + x^-2 + 1 = 0

Which is exactly what we wanted.

So
tan(4p/11) + 4sin(p/11) = __
Ö11

. QED