Finding Exact Value of Trig Function with Half-Angle Formula

[Chop235]

I am stuck on this problem:

It tells me to use the Half-Angle Formulas to find the exact value of the trig function.

sec((15*pi)/8)

I know I should convert that into the cos function and when I do I get:

cos(8/(15*pi))

But now I need to find cos(alpha/2) to be able to use the Half-Angle formula. I think what I can do is multiply it by 2 and then that would be my alpha, then I could just plug it in right? So then I would have:

cos((16/(15*pi))/2)

I am not sure if this is right so far or if I am even on the right track.

 

[verty]

As far as I can see:

sec(15*pi/8)
= sec(2pi - pi/8)
= sec(pi/8)

 

[HallsofIvy]

I am stuck on this problem:
It tells me to use the Half-Angle Formulas to find the exact value of the trig function.
sec((15*pi)/8)
I know I should convert that into the cos function and when I do I get:
cos(8/(15*pi))

NO! sec x= 1/(cos(x)) not cos(1/x)!

But now I need to find cos(alpha/2) to be able to use the Half-Angle formula. I think what I can do is multiply it by 2 and then that would be my alpha, then I could just plug it in right? So then I would have:
cos((16/(15*pi))/2)
I am not sure if this is right so far or if I am even on the right track.

Use the Half-Angle Formula alright but to go the other way!
sec(15pi/8)= 1/cos(15pi/8) so you just need to find cos(15pi/8)
That is $$cos(15pi/8)= \sqrt{\frac{1}{2}(1+ cos(15pi/4))}$$.
cos(15pi/4) is easy.