Simplify. \( \frac{\sin x}{1+\cos x}+\frac{1+\cos x}{\sin x} \) \( [?] \)

Calculate or simplify the expression \( \frac{\sin x}{1+\cos x}+\frac{1+\cos x}{\sin x} \). Simplify the expression by following steps: - step0: Solution: \(\frac{\sin\left(x\right)}{1+\cos\left(x\right)}+\frac{1+\cos\left(x\right)}{\sin\left(x\right)}\) - step1: Reduce fractions to a common denominator: \(\frac{\sin\left(x\right)\sin\left(x\right)}{\left(1+\cos\left(x\right)\right)\sin\left(x\right)}+\frac{\left(1+\cos\left(x\right)\right)\left(1+\cos\left(x\right)\right)}{\sin\left(x\right)\left(1+\cos\left(x\right)\right)}\) - step2: Rewrite the expression: \(\frac{\sin\left(x\right)\sin\left(x\right)}{\left(1+\cos\left(x\right)\right)\sin\left(x\right)}+\frac{\left(1+\cos\left(x\right)\right)\left(1+\cos\left(x\right)\right)}{\left(1+\cos\left(x\right)\right)\sin\left(x\right)}\) - step3: Transform the expression: \(\frac{\sin\left(x\right)\sin\left(x\right)+\left(1+\cos\left(x\right)\right)\left(1+\cos\left(x\right)\right)}{\left(1+\cos\left(x\right)\right)\sin\left(x\right)}\) - step4: Multiply the terms: \(\frac{\sin^{2}\left(x\right)+\left(1+\cos\left(x\right)\right)\left(1+\cos\left(x\right)\right)}{\left(1+\cos\left(x\right)\right)\sin\left(x\right)}\) - step5: Multiply the terms: \(\frac{\sin^{2}\left(x\right)+1+2\cos\left(x\right)+\cos^{2}\left(x\right)}{\left(1+\cos\left(x\right)\right)\sin\left(x\right)}\) - step6: Transform the expression: \(\frac{1+1+2\cos\left(x\right)}{\left(1+\cos\left(x\right)\right)\sin\left(x\right)}\) - step7: Add the numbers: \(\frac{2+2\cos\left(x\right)}{\left(1+\cos\left(x\right)\right)\sin\left(x\right)}\) - step8: Factor the expression: \(\frac{2\left(1+\cos\left(x\right)\right)}{\left(1+\cos\left(x\right)\right)\sin\left(x\right)}\) - step9: Reduce the fraction: \(\frac{2}{\sin\left(x\right)}\) - step10: Transform the expression: \(2\csc\left(x\right)\) The simplified expression is \(2\csc(x)\).