verify the identity.\n\\( \\frac{\\sin (9 x)-\\sin (x)}{\\cos (9 x)-\\cos (x)}=-\\cot (5 x) \\)\nstart with…

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

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

Answer

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))