Question 841560
All angles below are in degrees


sec(165) = sec(120+45)
={{{1/(cos(120+45))}}}


(*)Using the formula:
cos(a + b) = cos(a)*cos(b) - sin(a)*sin(b)


={{{1/(cos(120)*cos(45)-sin(120)*sin(45)))}}}
={{{1/(cos(180-60)*cos(45)-sin(180-60)*sin(45)))}}}
={{{1/(-cos(60)*cos(45)-sin(60)*sin(45)))}}}
={{{1/((-1/2)*(1/2)*sqrt(2)-(1/2)*sqrt(3)*(1/2)*sqrt(2))))}}}
={{{1/((-1/4)*sqrt(2)-(1/4)*sqrt(6))))}}}
={{{1/((-sqrt(2)-sqrt(6))/4)}}}
={{{4/((-sqrt(2)-sqrt(6)))}}}
={{{4/(-(sqrt(2)+sqrt(6)))}}}
={{{-4/(sqrt(2)+sqrt(6))}}}
={{{(-4/(sqrt(2)+sqrt(6)))*((sqrt(2)-sqrt(6))/(sqrt(2)-sqrt(6)))}}}


(*)Remember:
{{{(a+b)*(a-b)=a^2-b^2}}}


={{{-4(sqrt(2)-sqrt(6))/(2-6)}}}
={{{-4(sqrt(2)-sqrt(6))/-4}}}
={{{sqrt(2)-sqrt(6)}}}


so,
{{{sec(165)=sqrt(2)-sqrt(6)}}}