Deriving the exact value of $\sin \pi/12$ using double angle identity
Double angle identity – $\sin 2A = 2\sin A \cos A$
So, $\sin (2 \frac{\pi}{12}) = 2 \sin \frac{\pi}{12} \cos \frac{\pi}{12} $
$\sin^2 A + \cos^2 A = 1 $ so, $\cos \frac{\pi}{12} = \sqrt{1- \sin^2 \frac{\pi}{12}}$
$\sin (2 \frac{\pi}{12}) =2 \sin \frac{\pi}{12}\sqrt{1- \sin^2 \frac{\pi}{12}} $
I square both sides of the equation to get:
$(1/2)^2 = 4 \sin^2 \frac{\pi}{12} (1- \sin^2 \frac{\pi}{12})$
With this, I am moving away from finding the exact value of $\sin \pi/12$ which is $\frac{\sqrt{3}-1}{2\sqrt{2}}$
Best Answer
Let $x := \sin^2\frac{\pi}{12}$. Then your last equality is $$\frac{1}{4} = 4x(1-x) = 4x - 4x^2.$$ The solutions of this quadratic equation are $x_{1,2} = \frac{1}{2} \pm \frac{\sqrt{3}}{4}$. Clearly, $x$ is smaller than $\frac{1}{2}$, so we must have $x = \frac{1}{2} - \frac{\sqrt{3}}{4}$. Taking the square root gives $$\sin\frac{\pi}{12} = \frac{\sqrt{2 - \sqrt{3}}}{2}.$$ Now since $(\sqrt{3}-1)^2 = 4 - 2\sqrt{3} = 2(2 - \sqrt{3})$, we get $$\sin\frac{\pi}{12} = \frac{\sqrt{3}-1}{2\sqrt{2}}$$ as expected.