How solve this trigo system ?

[Gerson W. Barbosa]

From the first equation,

cos(y)=cos(x)-1/2

(sin(y))^2=1-(cos(x))^2+cos(x)-1/4

After squaring the second equation and replacing, we get

(cos(x))^4 - (cos(x))^3 - 7/4*(cos(x))^2 + cos(x) + 39/64 = 0

which can be solved by PROOT on the 50g:

[ 1 -1 '-7/4' 1 '39/64' ]

PROOT ->

cos(x) = [ -.411437827766 .911437827766 -1.04903810568 1.54903810568 ]

The first two roots appear to be

'(1-√7)/4' and '(1+√7)/4',

respectively.

Edited to fix the sign of one of the exact solutions.