this reply is a bit late but you might still be interested to know the solution and it's not easy .

sec 60 =2

cos 60=1/2

Do you know the triple angle formula ,

cos 3x =4cos^3 x -3cos x

Substitute x=20 ,

cos 3(20)=4cos^3 20- 3cos 20

1/2=4cos^3 20-3 cos 20

8cos^3 20-6cos 20-1=0

Solving this ,

cos 20 = 0.94 , cos 20= -0.77(undefined) , cos 20 = -0.17 (undefined) --- since cos cant be -ve in the 1st quadrant

sec 20 = 1/0.94

sec 20 = 1.064