1. The problem is to understand the expression \(\sin 5x\) and \(\sin^5 x\) and how they differ. 2. \(\sin 5x\) means the sine of \(5x\), which is the sine function applied to the angle \(5x\). 3. \(\sin^5 x\) means \((\sin x)^5\), which is the sine of \(x\) raised to the fifth power. 4. The formula for \(\sin 5x\) can be expanded using the multiple-angle identity: $$\sin 5x = 16 \sin^5 x - 20 \sin^3 x + 5 \sin x$$ 5. This shows that \(\sin 5x\) is a polynomial in \(\sin x\) of degree 5. 6. \(\sin^5 x\) is simply \(\sin x\) multiplied by itself 5 times, no angle multiplication involved. 7. Important rule: \(\sin^n x = (\sin x)^n\) means power of the sine value, while \(\sin nx\) means sine of the angle multiplied by \(n\). 8. To summarize: - \(\sin 5x\) is the sine of the angle \(5x\). - \(\sin^5 x\) is the sine of \(x\) raised to the fifth power. 9. Using the identity, you can express \(\sin 5x\) in terms of powers of \(\sin x\) as shown above. This clarifies the difference and relationship between \(\sin 5x\) and \(\sin^5 x\).