Question

How do you verify the identity ${\cos }^2\left(x\right)-{\sin }^2\left(x\right)=1-2{\sin }^2\left(x\right)$.

Answer 1

Verify the given identity:

Given expression: ${\cos }^2\left(x\right)-{\sin }^2\left(x\right)$.

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

$\Rightarrow {\cos }^2\left(x\right)=1-{\sin }^2\left(x\right)$.

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

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

Therefore, the given identity, ${\cos }^2\left(x\right)-{\sin }^2\left(x\right)=1-2{\sin }^2\left(x\right)$ is verified.