Trigonometry question: prove that 1/sin(10°) = -2 + 8cos2(10°)

[amine]

Trigonometry question: prove that 1/sin(10°) = -2 + 8cos2(10°)

How do you prove that
1/sin(10°) = -2 + 8cos²(10°) ?

 

[Deleted member 4993]

How do you prove that
1/sin(10°) = -2 + 8cos²(10°) ?

Calculator? .... just kidding

What are your thoughts?

Can you express sin(x) in terms of cos(x)?

 

[Dr.Peterson]

How do you prove that
1/sin(10°) = -2 + 8cos²(10°) ?

This is not true of all angles, so you have to start with something specific about 10°, and then try to use identities to obtain the desired fact.

I would start with the fact that sin(3*10°) = 1/2. Expand the left side using the triple angle identity (or, if you don't know that, the angle-sum and double-angle identities). This will end up giving you a cubic equation in the sine or cosine, which you fortunately don't have to solve. But you can show that it is equivalent to the fact you are trying to prove!

Have fun -- it's a nice challenge.