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Prove the trigonometric identity $\sec\left(x\right)=\frac{\sin\left(2x\right)}{\sin\left(x\right)}-\left(\frac{\cos\left(2x\right)}{\cos\left(x\right)}\right)$

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Worked example: Derivatives of sin(x) and cos(x) | Derivative rules | AP Calculus AB | Khan Academy

$\sec\left(x\right)=\frac{\sin\left(2x\right)}{\sin\left(x\right)}-\frac{\cos\left(2x\right)}{\cos\left(x\right)}$