SOLUTION: Simplify the expression by using a double-angle formula. 1-2sin^2(3π/5)

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Question 984661: Simplify the expression by using a double-angle formula.
1-2sin^2(3π/5)
Answer by ikleyn(52781) About Me  (Show Source):
You can put this solution on YOUR website!

There is the classic formula for the sines of the half argument:

sin%5E2%28alpha%2F2%29 = %281-cos%28alpha%29%29%2F2.

(see,  for example,  the lessons  FORMULAS FOR TRIGONOMETRIC FUNCTIONS  and  Trigonometric functions of half argument  in this site,  or any systematic textbook on trigonometry).

From this formula

cos%28alpha%29 = 1+-+2sin%5E2%28alpha%2F2%29.

By applying this formula to our case,  you will get

1+-+2sin%5E2%283pi%2F5%29 = cos%286pi%2F5%29 = -cos%28pi%2F5%29 = -cos(36°).

Answer.  1+-+2sin%5E2%283pi%2F5%29 = -cos%28pi%2F5%29 = -cos(36°).