How to prove $\\cos \\frac{2\\pi }{5}=\\frac{-1+\\sqrt{5}}{4}$?
I would like to find the apothem of a regular pentagon. It follows from
$$\\cos \\dfrac{2\\pi }{5}=\\dfrac{-1+\\sqrt{5}}{4}.$$
But how can this be proved (geometrically or trigonometrically)?
Since the solution involves much algebra, I'll only sketch the main steps/ideas of the solution here. First, use the multiple-angle formula (obtained by repeated use of the sum and double-angle formulas, etc):
Put and in the previous equation, then factor and eliminate x on both sides to get the following quadratic equation in the square of x: Solving for x, we get the value of sin pi/5. Finally, we obtain the desired value of cos pi/5 by solving for it in the equation (sin pi/5)^2 + (cos pi/5)^2 = 1.
and 
in the previous equation, then factor and eliminate x on both sides to get the following quadratic equation in the square of x: 
Solving for x, we get the value of sin pi/5. Finally, we obtain the desired value of cos pi/5 by solving for it in the equation (sin pi/5)^2 + (cos pi/5)^2 = 1.
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