Question

2sec(11)sec(19)-2cot(71)

Answer 1

I notice that
cos(30) = cos(11+19)
cos30 = cos11cos19 - sin11sin19
√3/2 + sin11sin19 = cos11cos19

and cot71 = cos71/sin71
= cos(60+11)/sin(60+11)
= (cos60cos11 - sin60sin11)/sin60cos11 + cos60sin11)
= (1/2 cos11 - √3/2 sin11)/(√3/2 cos11 + 1/2 sin11)
= (cos11 - √3sin11)/(√3cos11 + sin11)



2sec(11)sec(19)-2cot(71)
= 2/(cos11cos19) - 2cot71
= 2/(√3/2 + sin11sin19 ) - 2(cos11 - √3sin11)/(√3cos11 + sin11)

= 4/(√3 + 2sin11sin19) - 2(cos11 - √3sin11)/(√3cos11 + sin11)

At this point I don't see my way out of this quagmire, but I realize that 11+19 - 30, perhaps somebody else can see the light from here.