Questions: Verify the identity. sin(5x) - sin(x) / cos(5x) - cos(x) = -cot(3x) Start with the numerator of the left side and apply the appropriate formula of sum-to-product. sin(5x) - sin(x) = 2 sin((5x-x)/2) cos((5x+x)/2) (Do not simplify.) Now use the sum-to-product formula on the denominator of the left side. cos(5x) - cos(x) = (Do not simplify.)

Verify the identity.
sin(5x) - sin(x) / cos(5x) - cos(x) = -cot(3x)

Start with the numerator of the left side and apply the appropriate formula of sum-to-product.
sin(5x) - sin(x) = 2 sin((5x-x)/2) cos((5x+x)/2) (Do not simplify.)

Now use the sum-to-product formula on the denominator of the left side.
cos(5x) - cos(x) = (Do not simplify.)

Transcript text: Verify the identity. \[ \frac{\sin (5 x)-\sin (x)}{\cos (5 x)-\cos (x)}=-\cot (3 x) \] Start with the numerator of the left side and apply the appropriate formula of sum-to-product. \[ \sin (5 x)-\sin (x)=2 \sin \frac{5 x-x}{2} \cos \frac{5 x+x}{2} \text { (Do not simplify.) } \] Now use the sum-to-product formula on the denominator of the left side. \[ \cos (5 x)-\cos (x)= \] $\square$ (Do not simplify.)

Solution

Solution Steps

To verify the given trigonometric identity, we will use the sum-to-product formulas for both the numerator and the denominator. We will then simplify the resulting expressions and compare them to the right-hand side of the identity.

Solution Approach

Apply the sum-to-product formula to the numerator: $\sin(5x) - \sin(x)$.
Apply the sum-to-product formula to the denominator: $\cos(5x) - \cos(x)$.
Simplify the resulting expressions and verify if they match $-\cot(3x)$.

Step 1: Apply Sum-to-Product Formula to the Numerator

Given the numerator $\sin(5x) - \sin(x)$, we apply the sum-to-product formula: \[ \sin(5x) - \sin(x) = 2 \sin\left(\frac{5x - x}{2}\right) \cos\left(\frac{5x + x}{2}\right) \] Simplifying the arguments: \[ \sin(5x) - \sin(x) = 2 \sin(2x) \cos(3x) \]

Step 2: Apply Sum-to-Product Formula to the Denominator

Given the denominator $\cos(5x) - \cos(x)$, we apply the sum-to-product formula: \[ \cos(5x) - \cos(x) = -2 \sin\left(\frac{5x + x}{2}\right) \sin\left(\frac{5x - x}{2}\right) \] Simplifying the arguments: \[ \cos(5x) - \cos(x) = -2 \sin(3x) \sin(2x) \]

Step 3: Simplify the Left-Hand Side

Using the simplified forms of the numerator and the denominator: \[ \frac{\sin(5x) - \sin(x)}{\cos(5x) - \cos(x)} = \frac{2 \sin(2x) \cos(3x)}{-2 \sin(2x) \sin(3x)} \] Canceling the common factor $2 \sin(2x)$: \[ \frac{\sin(2x) \cos(3x)}{-\sin(2x) \sin(3x)} = -\frac{\cos(3x)}{\sin(3x)} \] This simplifies to: \[ -\cot(3x) \]

Step 4: Verify the Identity

The left-hand side simplifies to: \[ -\cot(3x) \] which matches the right-hand side of the given identity.