Question

How do you use the trigonometric identity $\cos 2x={\cos }^{2}x-{\sin }^{2}x$ to verify that $\cos 2x=2{\cos }^{2}x-1$.

Answer 1

Verify the given identity:

Given identity: $\cos 2x={\cos }^{2}x-{\sin }^{2}x$.

It is known that, ${\cos }^{2}x+{\sin }^{2}x=1$

$\Rightarrow {\sin }^{2}x=1-{\cos }^{2}x$.

Substitute ${\sin }^{2}x$ with $\left(1-{\cos }^{2}x\right)$ in given identity

$$\begin{gathered} \Rightarrow \cos 2x={\cos }^{2}x-\left(1-{\cos }^{2}x\right) \\ \Rightarrow \cos 2x={\cos }^{2}x-1+{\cos }^{2}x \\ \Rightarrow \cos 2x=2{\cos }^{2}x-1 \end{gathered}$$

Therefore, the given identity, $\cos 2x=2{\cos }^{2}x-1$ is verified.