How to use the trigonometric identity $\cos(2A)=1-2\sin^2(A)$ to show that $\sin\left(\frac{\pi}{12}\right)=\sqrt{\frac{1}{2} - \frac{\sqrt{3}}{4}}$

Question

Question: Use the trigonometric identity $\cos(2A)=1-2\sin^2(A)$ to show that $$\sin\left(\frac{\pi}{12}\right)=\sqrt{\frac{1}{2} - \frac{\sqrt{3}}{4}}$$

What are good strategies to figure out this question in Particular?

Answer 1

Use the trigonemetric identity, we get $\sin{A}=\pm\sqrt{\frac{1-\cos{2A}}{2}}$

Substitute $A=\frac{\pi}{12}$ in it, we can easily find that $\sin A>0$. Therefore $\sin{\frac{\pi}{12}}=\sqrt{\frac{1-\cos{\frac{\pi}{6}}}{2}}=\sqrt{\frac{1-\frac{\sqrt{3}}{2}}{2}}=\sqrt{\frac{1}{2}-\frac{\sqrt{3}}{4}}$

Answer 2

$$\cos(2A)=1-2\sin^2(A)$$

$$\cos(\pi /6)=1-2\sin^2(\pi /{12})$$

$$1-2\sin^2(\pi /{12})=\frac {\sqrt 3}{2}$$

$$2\sin^2(\pi /{12})=1-\frac {\sqrt 3}{2}$$ $$ \sin^2(\pi /{12})=\frac {1}{2}-\frac {\sqrt 3}{4}$$ $$\sin(\pi /{12})=\sqrt {\frac {1}{2}-\frac {\sqrt 3}{4}}$$