Question

Given that sin (pi/10)=(sqrt(5)-1)/4, use double-angle formulas to find an exact expression for sin(pi/5).

Answer 1

Just for easier typing I will use
π/10 radians = 18°
π/5 radians = 36°

The formula you want is
cos 36° = 1 - 2sin^2 18°
= 1 - 2(√5-1)^2/16
= 1 - (√5 - 1)^2/8
= (8 - (5 - 2√5 + 1))/8
= (1+√5)/4

but we wanted sin 36° or sin(π/5)

sin^2 36° + cos^2 36° = 1
sin^2 36 = 1 - (1+√5)^2/16
= (16 - (1 + 2√5 + 5))/16
= (5-√5)/8

sin 36° = sin (π/5) = √[(5-√5)/8]