\( \frac { 5 \operatorname { cosex } } { \frac { ( 2 x - 3 ) } { 4 x } } - \frac { ( 3 x - 2 x } { 3 } \)

Simplify the expression by following steps: - step0: Solution: \(\frac{5cosex}{\left(\frac{\left(2x-3\right)}{4x}\right)}-\left(\frac{\left(3x-2x\right)}{3}\right)\) - step1: Remove the parentheses: \(\frac{5cosex}{\frac{2x-3}{4x}}-\left(\frac{3x-2x}{3}\right)\) - step2: Subtract the terms: \(\frac{5cosex}{\frac{2x-3}{4x}}-\left(\frac{x}{3}\right)\) - step3: Remove the parentheses: \(\frac{5cosex}{\frac{2x-3}{4x}}-\frac{x}{3}\) - step4: Multiply the numbers: \(\frac{5ecosx}{\frac{2x-3}{4x}}-\frac{x}{3}\) - step5: Divide the terms: \(\frac{20ecosx^{2}}{2x-3}-\frac{x}{3}\) - step6: Reduce fractions to a common denominator: \(\frac{20ecosx^{2}\times 3}{\left(2x-3\right)\times 3}-\frac{x\left(2x-3\right)}{3\left(2x-3\right)}\) - step7: Reorder the terms: \(\frac{20ecosx^{2}\times 3}{3\left(2x-3\right)}-\frac{x\left(2x-3\right)}{3\left(2x-3\right)}\) - step8: Transform the expression: \(\frac{20ecosx^{2}\times 3-x\left(2x-3\right)}{3\left(2x-3\right)}\) - step9: Multiply the terms: \(\frac{60ecosx^{2}-x\left(2x-3\right)}{3\left(2x-3\right)}\) - step10: Multiply the terms: \(\frac{60ecosx^{2}-\left(2x^{2}-3x\right)}{3\left(2x-3\right)}\) - step11: Remove the parentheses: \(\frac{60ecosx^{2}-2x^{2}+3x}{3\left(2x-3\right)}\) - step12: Simplify: \(\frac{60ecosx^{2}-2x^{2}+3x}{6x-9}\) Let's simplify the expression step by step: 1. **Original Expression**: \[ \frac{5 \operatorname{cosex}}{\frac{(2x - 3)}{4x}} - \frac{(3x - 2x)}{3} \] 2. **Simplifying the First Term**: The first term can be rewritten as: \[ \frac{5 \operatorname{cosex}}{\frac{(2x - 3)}{4x}} = 5 \operatorname{cosex} \cdot \frac{4x}{(2x - 3)} = \frac{20x \operatorname{cosex}}{(2x - 3)} \] 3. **Simplifying the Second Term**: The second term simplifies to: \[ \frac{(3x - 2x)}{3} = \frac{x}{3} \] 4. **Combining the Terms**: Now we combine both terms: \[ \frac{20x \operatorname{cosex}}{(2x - 3)} - \frac{x}{3} \] 5. **Finding a Common Denominator**: The common denominator for the two fractions is \(3(2x - 3)\). Thus, we rewrite the expression: \[ \frac{20x \operatorname{cosex} \cdot 3}{3(2x - 3)} - \frac{x(2x - 3)}{3(2x - 3)} \] 6. **Combining the Fractions**: This gives us: \[ \frac{60x \operatorname{cosex} - (2x^2 - 3x)}{3(2x - 3)} \] 7. **Final Simplified Expression**: The final simplified expression is: \[ \frac{60x \operatorname{cosex} - 2x^2 + 3x}{3(2x - 3)} \] Thus, the simplified result is: \[ \frac{60 \operatorname{cosex} x^2 - 2x^2 + 3x}{6x - 9} \]