Answer
The expression simplifies to \(\frac{411\cos(5x) + 211\cos(x)}{200}\).
Solution
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 \(\frac{411\cos(5x) + 211\cos(x)}{200}\).
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