Final Answer
Step-by-step Solution
Problem to solve:
Solving method
Multiplying the fraction by $-1$
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 $\sin\left(x\right)$
Combine all terms into a single fraction with $\cos\left(x\right)$ as common denominator
When multiplying two powers that have the same base ($\cos\left(x\right)$), you can add the exponents
Apply the trigonometric identity: $\cos\left(2x\right)$$=2\cos\left(x\right)^2-1$
Solve the product $-(2\cos\left(x\right)^2-1)$
Applying the trigonometric identity: $\displaystyle\sec\left(\theta\right)=\frac{1}{\cos\left(\theta\right)}$
Since both sides of the equality are equal, we have proven the identity