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I want to expand $\cos\left( \dfrac{2\pi}{n} \right)$ for $n=5,~17,~ 257, 65537$ to radicals, for example: $\cos\left( \dfrac{2\pi}{5} \right) =\frac14 \sqrt{5} -\frac14$

Is this a possible task?

x = var('x')
test = (x+1)^2 # test
test.expand() # works :)

term = cos(2*pi/5)
term.expand() # works :)

term = cos(2*pi/17)
term.expand() # works not :(

PS: I tried

cos(2*pi/17).trig_simplify()

cos(2*pi/17).trig_expand()

as well.

I want to expand $\cos\left( \dfrac{2\pi}{n} \right)$ for $n=5,~17,~ 257, 65537$ to radicals, for example: $\cos\left( \dfrac{2\pi}{5} \right) =\frac14 \sqrt{5} -\frac14$

Is this a possible task?

x = var('x')
test = (x+1)^2 # test
test.expand() # works :)

term = cos(2*pi/5)
term.expand() # works :)

term = cos(2*pi/17)
term.expand() # works not :(

PS: I tried

cos(2*pi/17).trig_simplify()

cos(2*pi/17).trig_expand()

as well.

I want to expand $\cos\left( \dfrac{2\pi}{n} \right)$ for $n=5,~17,~ 257, 65537$ to radicals, for example: $\cos\left( \dfrac{2\pi}{5} \right) =\frac14 \sqrt{5} -\frac14$

Is this a possible task?

x = var('x')
test = (x+1)^2 # test
test.expand() # works :)

term = cos(2*pi/5)
term.expand() # works :)

term = cos(2*pi/17)
term.expand() # works not :(

PS: I tried

cos(2*pi/17).trig_simplify()

cos(2*pi/17).trig_expand()

as well.

I want to expand $\cos\left( \dfrac{2\pi}{n} \right)$ for $n=5,~17,~ 257, 65537$ to radicals, for example: $\cos\left( \dfrac{2\pi}{5} \right) =\frac14 \sqrt{5} -\frac14$

Is this a possible task?

x = var('x')
test = (x+1)^2 # test
test.expand() # works :)

term = cos(2*pi/5)
term.expand() # works :)

term = cos(2*pi/17)
term.expand() # works not :(

PS: I tried

cos(2*pi/17).trig_simplify()

cos(2*pi/17).trig_expand()

as well.

I want to expand $\cos\left( \dfrac{2\pi}{n} \right)$ for $n=5,~17,~ 257, 65537$ to radicals, for example: $\cos\left( \dfrac{2\pi}{5} \right) =\frac14 \sqrt{5} -\frac14$

BTW: These are Fermat prime numbers, see Exact trigonometric values.

Is this a possible task?

x = var('x')
test = (x+1)^2 # test
test.expand() # works :)

term = cos(2*pi/5)
term.expand() # works :)

term = cos(2*pi/17)
term.expand() # works not :(

PS: I tried

cos(2*pi/17).trig_simplify()

cos(2*pi/17).trig_expand()

as well. well.

I want to expand $\cos\left( \dfrac{2\pi}{n} \right)$ for $n=5,~17,~ 257, 65537$ to radicals, for example: $\cos\left( \dfrac{2\pi}{5} \right) =\frac14 \sqrt{5} -\frac14$

BTW: These $n$'s are Fermat prime numbers, see Exact trigonometric values.

Is this a possible task?

x = var('x')
test = (x+1)^2 # test
test.expand() # works :)

term = cos(2*pi/5)
term.expand() # works :)

term = cos(2*pi/17)
term.expand() # works not :(

PS: I tried

cos(2*pi/17).trig_simplify()

cos(2*pi/17).trig_expand()

as well.

I want to expand $\cos\left( \dfrac{2\pi}{n} \right)$ for $n=5,~17,~ 257, 65537$ to radicals, for example: $\cos\left( \dfrac{2\pi}{5} \right) =\frac14 \sqrt{5} -\frac14$

BTW: These $n$'s $n$ are Fermat prime numbers, see Exact trigonometric values.

Is this a possible task?

x = var('x')
test = (x+1)^2 # test
test.expand() # works :)

term = cos(2*pi/5)
term.expand() # works :)

term = cos(2*pi/17)
term.expand() # works not :(

PS: I tried

cos(2*pi/17).trig_simplify()

cos(2*pi/17).trig_expand()

as well.