Questions: Verify the identity. [ fracsin (3 x)-sin (5 x)cos (3 x)-cos (5 x)=-cot (4 x) ] Start with the numerator of the left side and apply (sin (3 x)-sin (5 x)=).

Solution Steps

To verify the given trigonometric identity, we need to start by expressing the numerator \(\sin(3x) - \sin(5x)\) using trigonometric identities. Specifically, we can use the sum-to-product identities for sine and cosine.

1. Use the sum-to-product identities to express \(\sin(3x) - \sin(5x)\).
2. Similarly, use the sum-to-product identities to express \(\cos(3x) - \cos(5x)\).
3. Simplify the resulting expressions and verify if the left-hand side equals \(-\cot(4x)\).

We start with the expression for the numerator: \[ \sin(3x) - \sin(5x) \] Using the sum-to-product identity, we can express this as: \[ \sin(3x) - \sin(5x = 2 \cos\left(\frac{3x + 5x}{2}\right) \sin\left(\frac{5x - 3x}{2}\right) = 2 \cos(4x) \sin(x) \]

Next, we consider the denominator: \[ \cos(3x) - \cos(5x) \] Using the sum-to-product identity, we can express this as: \[ \cos(3x) - \cos(5x) = -2 \sin\left(\frac{3x + 5x}{2}\right) \sin\left(\frac{5x - 3x}{2}\right) = -2 \sin(4x) \sin(x) \]

Now we can substitute the simplified numerator and denominator into the left-hand side of the identity: \[ \frac{\sin(3x) - \sin(5x)}{\cos(3x) - \cos(5x)} = \frac{2 \cos(4x) \sin(x)}{-2 \sin(4x) \sin(x)} \] This simplifies to: \[ -\frac{\cos(4x)}{\sin(4x)} = -\cot(4x) \]

We find that the left-hand side simplifies to: \[ -\cot(4x) \] which matches the right-hand side of the original identity. Therefore, the identity is verified.

Final Answer