Simplify into a single trigonometric ratio. \[ \sqrt{\frac{\frac{1}{2} \sin 2 x}{\tan \left(540^{\circ}+x\right)\left(\frac{1}{\cos ^{2} x}-\tan ^{2} x\right)}} \]

Simplify the expression by following steps: - step0: Solution: \(\tan\left(540^{\circ}+x\right)\) - step1: Rewrite the expression: \(\tan\left(x\right)\) Calculate or simplify the expression \( \frac{1}{2} \sin 2 x \). Simplify the expression by following steps: - step0: Solution: \(\frac{1}{2}\sin\left(2x\right)\) Calculate or simplify the expression \( \frac{1}{\cos ^{2} x}-\tan ^{2} x \). Simplify the expression by following steps: - step0: Solution: \(\frac{1}{\cos^{2}\left(x\right)}-\tan^{2}\left(x\right)\) - step1: Reduce fractions to a common denominator: \(\frac{1}{\cos^{2}\left(x\right)}-\frac{\tan^{2}\left(x\right)\cos^{2}\left(x\right)}{\cos^{2}\left(x\right)}\) - step2: Transform the expression: \(\frac{1-\tan^{2}\left(x\right)\cos^{2}\left(x\right)}{\cos^{2}\left(x\right)}\) - step3: Transform the expression: \(-\frac{\left(\tan\left(x\right)\cos\left(x\right)+1\right)\left(\tan\left(x\right)\cos\left(x\right)-1\right)}{\cos^{2}\left(x\right)}\) - step4: Transform the expression: \(-\frac{\left(\sin\left(x\right)+1\right)\left(\sin\left(x\right)-1\right)}{\cos^{2}\left(x\right)}\) - step5: Transform the expression: \(-\left(-1\right)\) - step6: Calculate: \(1\) Calculate or simplify the expression \( \sqrt{\frac{\frac{1}{2}\sin(2x)}{\tan(x)(1)}} \). Simplify the expression by following steps: - step0: Solution: \(\sqrt{\frac{\frac{1}{2}\sin\left(2x\right)}{\tan\left(x\right)\times 1}}\) - step1: Multiply: \(\sqrt{\frac{\frac{1}{2}\sin\left(2x\right)}{\tan\left(x\right)}}\) - step2: Divide the terms: \(\sqrt{\frac{\sin\left(2x\right)}{2\tan\left(x\right)}}\) - step3: Use the properties of radicals: \(\frac{\sqrt{\sin\left(2x\right)}}{\sqrt{2\tan\left(x\right)}}\) - step4: Transform the expression: \(\frac{\sqrt{\sin\left(2x\right)}}{\frac{\sqrt{2\sin\left(x\right)}}{\sqrt{\cos\left(x\right)}}}\) - step5: Multiply by the reciprocal: \(\sqrt{\sin\left(2x\right)}\times \frac{\sqrt{\cos\left(x\right)}}{\sqrt{2\sin\left(x\right)}}\) - step6: Multiply the terms: \(\frac{\sqrt{\sin\left(2x\right)}\times \sqrt{\cos\left(x\right)}}{\sqrt{2\sin\left(x\right)}}\) - step7: Multiply the terms: \(\frac{\sqrt{\sin\left(2x\right)\cos\left(x\right)}}{\sqrt{2\sin\left(x\right)}}\) - step8: Transform the expression: \(\frac{\sqrt{2\sin\left(x\right)}\times \cos\left(x\right)}{\sqrt{2\sin\left(x\right)}}\) - step9: Reduce the fraction: \(\cos\left(x\right)\) To simplify the expression \[ \sqrt{\frac{\frac{1}{2} \sin 2 x}{\tan \left(540^{\circ}+x\right)\left(\frac{1}{\cos ^{2} x}-\tan ^{2} x\right)}} \] we follow these steps: 1. **Simplify \(\frac{1}{2} \sin 2x\)**: - This remains as \(\frac{1}{2} \sin 2x\). 2. **Simplify \(\tan(540^{\circ} + x)\)**: - Since \(\tan\) has a period of \(180^{\circ}\), we can reduce \(540^{\circ}\) to \(540^{\circ} - 3 \times 180^{\circ} = 540^{\circ} - 540^{\circ} = 0^{\circ}\). - Thus, \(\tan(540^{\circ} + x) = \tan(x)\). 3. **Simplify \(\frac{1}{\cos^2 x} - \tan^2 x\)**: - Using the identity \(\tan^2 x = \frac{\sin^2 x}{\cos^2 x}\), we have: \[ \frac{1}{\cos^2 x} - \tan^2 x = \frac{1}{\cos^2 x} - \frac{\sin^2 x}{\cos^2 x} = \frac{1 - \sin^2 x}{\cos^2 x} = \frac{\cos^2 x}{\cos^2 x} = 1. \] 4. **Substituting back into the original expression**: - Now we substitute these simplifications back into the expression: \[ \sqrt{\frac{\frac{1}{2} \sin 2x}{\tan(x) \cdot 1}} = \sqrt{\frac{\frac{1}{2} \sin 2x}{\tan(x)}}. \] 5. **Using the identity \(\sin 2x = 2 \sin x \cos x\)**: - We can rewrite \(\sin 2x\) as \(2 \sin x \cos x\): \[ \sqrt{\frac{\frac{1}{2} \cdot 2 \sin x \cos x}{\tan(x)}} = \sqrt{\frac{\sin x \cos x}{\tan(x)}}. \] 6. **Substituting \(\tan(x) = \frac{\sin x}{\cos x}\)**: - Thus, we have: \[ \sqrt{\frac{\sin x \cos x}{\frac{\sin x}{\cos x}}} = \sqrt{\cos^2 x} = \cos x. \] Therefore, the simplified expression is \[ \cos x. \]