Question

How do you find the exact value of `cos68` using the sum and difference, double angle or half angle formulas?

Answer 1

This angle is not constructable, so its cosine will not be related to any sum and difference formulas made out of all constructible angles . This means there won't be any closed formed expression combining integers using addition, subtraction, multiplication, division and square roots. There is a way to get something using nth roots of complex numbers but that's not really the sum and difference formulas so I'll skip that here.

Just to pedantically answer the question,

`cos 68^circ = cos 60^circ cos 8^circ - sin 60^circ sin 8^circ`

That's exact, though with trig functions of `8^circ` it's not that useful.