Starting from the right-hand side (RHS) of the identity

Using the sine double-angle identity: $\sin\left(2\theta\right)=2\sin\left(\theta\right)\cos\left(\theta\right)$

Simplify the fraction $\frac{2\sin\left(x\right)\cos\left(x\right)}{\sin\left(x\right)}$ by $\sin\left(x\right)$

Apply the trigonometric identity: $\cos\left(2\theta \right)$$=2\cos\left(\theta \right)^2-1$

Cancel like terms $2\cos\left(x\right)^2$ and $-2\cos\left(x\right)^2$

Applying the trigonometric identity: $\displaystyle\sec\left(\theta\right)=\frac{1}{\cos\left(\theta\right)}$

Since we have reached the expression of our goal, we have proven the identity