"Completely" is a fuzzy goal in trig, as we shall see.
First, the point of this problem is to recognize the sine form of the cosine double angle formula:
#cos(2 theta) = cos(theta + theta) = cos theta cos theta - sin theta sin theta ##= cos^2 theta - sin ^2 theta = (1- sin^2 theta) - sin^2 theta #
#cos(2 theta) = 1 - 2 sin ^2 theta#
Writing this for #theta=20^circ#,
#cos( 2 (20^circ) ) = 1 - 2 sin^2 20^circ#
#cos 40^circ= 1 - 2 sin^2 20^circ#
Presumably #cos 40^circ# is the result "simplified completely."
That's the answer. Monzur suggests I give a caveat before the next part. It's totally optional; please keep reading if you want to know more about #cos 40^circ# and stop reading if you don't.
#cos 40^circ# is simplified completely, because there's not really any better expression we can write down for it than #cos 40^circ#. #40^circ# is not constructible with a straightedge and compass. That means its trig functions are not the result of integers composed by addition, subtraction, multiplication and division and square roots.
#cos 40^circ# is actually the root of a polynomial equation with integer coefficients. #theta=40^circ# satisfies the equation #cos(44 theta)=-cos(46 theta)#. If #x = cos theta,# that's #T_{44}(x) = -T_{46}(x),# where the #T#s are the Chebyshev polynomials of the first kind. Instead of the double and triple angle formulas, they're the 44 times and 46 times angle formulas.
So #cos(40^circ)# is one of the forty-six roots of:
#8796093022208 x^44 - 96757023244288 x^42 + 495879744126976 x^40 - 1572301627719680 x^38 + 3454150138396672 x^36 - 5579780992794624 x^34 + 6864598984556544 x^32 - 6573052309536768 x^30 + 4964023879598080 x^28 - 2978414327758848 x^26 + 1423506847825920 x^24 - 541167892561920 x^22 + 162773155184640 x^20 - 38370843033600 x^18 + 6988974981120 x^16 - 963996549120 x^14 + 97905899520 x^12 - 7038986240 x^10 + 338412800 x^8 - 9974272 x^6 + 155848 x^4 - 968 x^2 + 1 = - ( 35184372088832 x^46 - 404620279021568 x^44 + 2174833999740928 x^42 - 7257876254949376 x^40 + 16848641306132480 x^38 - 28889255702953984 x^36 + 37917148110127104 x^34 - 38958828003262464 x^32 + 31782201792135168 x^30 - 20758645314682880 x^28 + 10898288790208512 x^26 - 4599927086776320 x^24 + 1555857691115520 x^22 - 418884762992640 x^20 + 88826010009600 x^18 - 14613311324160 x^16 + 1826663915520 x^14 - 168586629120 x^12 + 11038410240 x^10 - 484140800 x^8 + 13034560 x^6 - 186208 x^4 + 1058 x^2 - 1
) #
That's not simple at all.
