Final Answer
Step-by-step explanation
Problem to solve:
Choose the 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 $2\cos\left(x\right)+\frac{-\cos\left(2x\right)}{\cos\left(x\right)}$ in a single fraction
When multiplying two powers that have the same base ($\cos\left(x\right)$), you can add the exponents
Applying an identity of double-angle cosine: $\cos\left(2\theta\right)=1-2\sin\left(\theta\right)^2$
Solve the product $-(1-2\sin\left(x\right)^2)$
Applying the pythagorean identity: $\sin^2\left(\theta\right)+\cos^2\left(\theta\right)=1$
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