trigonometry Addition Formulae (iv)

I have worked out how to do it, just I need to know how to solve it without calculator.

Find the exact value of:

2 cos²15 - 1

this is what i did:

2 cos²15 - 1 **so**

= cos²15 + sin²15 = 1 **side note:** sin²15 = 1 - cos²15 **hence:** 2 cos²15 - 1

then i did:

**2 cos²15 - 1** __would initially be:__ **cos²15 + sin²15**

= cos(15 + 15)

= cos 15 x cos 15 - sin 15 x sin 15

= root 3 / 2

What i wanna know is, i could not of done this without a calculator...

What is the alternative method?

(sorry if i made this very confusing)