A quadratic equation whose roots are $\sin^{2} 18^{\circ}, \cos^{2} 36^{\circ}$ are
- $16x^{2}-12x+1=0$
- $x^{2}-12x+1=0$
- $16x^{2}-12x-1=0$
-
none of these
sin 18 = (sqrt(5)-1)/4, so sin^2 18 = (5+1-2sqrt(5))/16 = (6-2sqrt(5))/16 = (3-sqrt(5))/8. cos 36 = (sqrt(5)+1)/4, so cos^2 36 = (5+1+2sqrt(5))/16 = (6+2sqrt(5))/16 = (3+sqrt(5))/8. Sum of roots = 6/8 = 3/4. Product of roots = (9-5)/64 = 4/64 = 1/16. Equation: x^2 - (3/4)x + 1/16 = 0, or 16x^2 - 12x + 1 = 0.
The value of sin^2(18 degrees) is found using the identity for the golden ratio to be (3 - sqrt(5))/8, and the value of cos^2(36 degrees) is (1 + cos(72 degrees))/2 = (3 + sqrt(5))/8. The sum of these two roots is 6/8 = 3/4 and their product is (9 - 5)/64 = 4/64 = 1/16. Forming the quadratic equation x^2 - (3/4)x + 1/16 = 0 and multiplying by 16 yields the equation 16x^2 - 12x + 1 = 0.