Compute cos(pi/5) based on cos(pi/10) using the half angle formula?

Question

The half angula formula says:
cos(A/2) = ± √( [1 + cos(A)]/2 )

cos(pi/10) = (1/4)√(10 + 2√5)

cos(pi/5) = (√5+1)/4

Use the half angle formula (and any other trigonometric identities you would like to use) to derive cos(pi/5) from cos(pi/10) using the above values.

References
* http://en.wikipedia.org/wiki/Trigonometric_identity
* http://en.wikipedia.org/wiki/Exact_trigonometric_constants

Answer 1

Half-angle formula is:
cos(A/2) = + sqrt {[1 + cos(A)] / 2}
for A/2 in quadrant 1.

Rearranging gives:
cos(A) = 2cos^2(A/2) - 1

Letting A = pi/5, then A/2 = pi/10.

Substituting for A and cos(A/2) in the equation gives:

cos(pi/5) = 2*{sqrt[10 + 2*sqrt(5)] / 4}^2 - 1

= 2*[10 + 2*sqrt(5)] / 16 - 1

= [10 + 2 * sqrt(5)] / 8 - 1

= [5 + sqrt(5)] / 4 - 1

= [sqrt(5) + 1] / 4, as required.