Question

verify the identity.
\\( \frac{\sin (9 x)-\sin (x)}{\cos (9 x)-\cos (x)}=-\cot (5 x) \\)
start with the numerator of the left side and apply the appropriate formula of sum - to - product.
\\( \sin (9 x)-\sin (x)=\square \\) (do not simplify )

Explanation:

Step1: Apply the sum - to - product formula for sine

The sum - to - product formula for \(\sin A-\sin B\) is \(2\cos\frac{A + B}{2}\sin\frac{A - B}{2}\).
Here \(A = 9x\) and \(B=x\), so \(\sin(9x)-\sin(x)=2\cos\frac{9x + x}{2}\sin\frac{9x - x}{2}\)

Step2: Simplify the arguments of cosine and sine

\(\frac{9x + x}{2}=\frac{10x}{2} = 5x\) and \(\frac{9x - x}{2}=\frac{8x}{2}=4x\)
So \(\sin(9x)-\sin(x)=2\cos(5x)\sin(4x)\)

Answer:

\(2\cos(5x)\sin(4x)\)