Final Answer
true
Step-by-step Solution
Problem to solve:
$\sec\left(x\right)=\frac{\sin\left(2x\right)}{\sin\left(x\right)}-\frac{\cos\left(2x\right)}{\cos\left(x\right)}$
Choose the solving method
1
Using the sine double-angle identity: $\sin\left(2\theta\right)=2\sin\left(\theta\right)\cos\left(\theta\right)$
$\sec\left(x\right)=\frac{2\sin\left(x\right)\cos\left(x\right)}{\sin\left(x\right)}+\frac{-\cos\left(2x\right)}{\cos\left(x\right)}$
2
Simplify the fraction $\frac{2\sin\left(x\right)\cos\left(x\right)}{\sin\left(x\right)}$ by $\sin\left(x\right)$
$\sec\left(x\right)=2\cos\left(x\right)+\frac{-\cos\left(2x\right)}{\cos\left(x\right)}$
3
Combine all terms into a single fraction with $\cos\left(x\right)$ as common denominator
$\sec\left(x\right)=\frac{2\cos\left(x\right)\cos\left(x\right)-\cos\left(2x\right)}{\cos\left(x\right)}$
4
When multiplying two powers that have the same base ($\cos\left(x\right)$), you can add the exponents
$\sec\left(x\right)=\frac{2\cos\left(x\right)^2-\cos\left(2x\right)}{\cos\left(x\right)}$
5
Apply the trigonometric identity: $2\cos\left(x\right)^2$$=\cos\left(2x\right)+1$
$\sec\left(x\right)=\frac{1}{\cos\left(x\right)}$
6
Applying the trigonometric identity: $\displaystyle\sec\left(\theta\right)=\frac{1}{\cos\left(\theta\right)}$
$\sec\left(x\right)=\sec\left(x\right)$
7
Since both sides of the equality are equal, we have proven the identity
true
Final Answer
true