Express the following as single trigonometry ratio: \( 3.11 \cos 2 x \cdot \cos 3 x-\sin 2 x \cdot \sin 3 x \)

Calculate or simplify the expression \( 3.11 \cos(2x) \cdot \cos(3x)-\sin(2x) \cdot \sin(3x) \). Simplify the expression by following steps: - step0: Solution: \(3.11\cos\left(2x\right)\cos\left(3x\right)-\sin\left(2x\right)\sin\left(3x\right)\) - step1: Transform the expression: \(\frac{311}{100}\cos\left(3x\right)\cos\left(2x\right)-\sin\left(3x\right)\sin\left(2x\right)\) - step2: Transform the expression: \(\frac{311}{100}\cos\left(3x\right)\left(\cos^{2}\left(x\right)-\sin^{2}\left(x\right)\right)-\sin\left(3x\right)\sin\left(2x\right)\) - step3: Transform the expression: \(\frac{311}{100}\cos\left(3x\right)\left(\cos^{2}\left(x\right)-\sin^{2}\left(x\right)\right)-2\sin\left(3x\right)\cos\left(x\right)\sin\left(x\right)\) - step4: Transform the expression: \(\frac{311}{100}\cos\left(3x\right)\left(1-2\sin^{2}\left(x\right)\right)-2\sin\left(3x\right)\cos\left(x\right)\sin\left(x\right)\) - step5: Transform the expression: \(\frac{311}{100}\cos\left(3x\right)-\frac{311}{50}\sin^{2}\left(x\right)\cos\left(3x\right)-2\sin\left(x\right)\sin\left(3x\right)\cos\left(x\right)\) - step6: Transform the expression: \(\frac{311}{100}\cos\left(3x\right)-\frac{311}{50}\cos\left(3x\right)\left(1-\cos^{2}\left(x\right)\right)-2\sin\left(x\right)\sin\left(3x\right)\cos\left(x\right)\) - step7: Transform the expression: \(\frac{311}{100}\cos\left(3x\right)-\frac{311\cos\left(3x\right)\left(1-\cos\left(2x\right)\right)}{100}-2\sin\left(x\right)\sin\left(3x\right)\cos\left(x\right)\) - step8: Subtract the terms: \(\frac{311\cos\left(2x\right)\cos\left(3x\right)}{100}-2\sin\left(x\right)\sin\left(3x\right)\cos\left(x\right)\) - step9: Reduce fractions to a common denominator: \(\frac{311\cos\left(2x\right)\cos\left(3x\right)}{100}-\frac{2\sin\left(x\right)\sin\left(3x\right)\cos\left(x\right)\times 100}{100}\) - step10: Transform the expression: \(\frac{311\cos\left(2x\right)\cos\left(3x\right)-2\sin\left(x\right)\sin\left(3x\right)\cos\left(x\right)\times 100}{100}\) - step11: Multiply the terms: \(\frac{311\cos\left(2x\right)\cos\left(3x\right)-200\sin\left(x\right)\sin\left(3x\right)\cos\left(x\right)}{100}\) - step12: Transform the expression: \(\frac{\frac{411\cos\left(5x\right)+211\cos\left(x\right)}{2}}{100}\) - step13: Multiply by the reciprocal: \(\frac{411\cos\left(5x\right)+211\cos\left(x\right)}{2}\times \frac{1}{100}\) - step14: Multiply the terms: \(\frac{411\cos\left(5x\right)+211\cos\left(x\right)}{2\times 100}\) - step15: Multiply the terms: \(\frac{411\cos\left(5x\right)+211\cos\left(x\right)}{200}\) The expression \(3.11 \cos 2x \cdot \cos 3x - \sin 2x \cdot \sin 3x\) can be simplified to \(\frac{411\cos(5x) + 211\cos(x)}{200}\).