verify the identity.\n\\frac{\\sin(7x)-\\sin(x)}{\\cos(7x)-\\cos(x)}=-\\cot(4x)\n\nstart with the numerator…

verify the identity.\n\\frac{\\sin(7x)-\\sin(x)}{\\cos(7x)-\\cos(x)}=-\\cot(4x)\n\nstart with the numerator of the left side and apply the appropriate formula of sum - to - product.\n\\sin(7x)-\\sin(x)=\\square (do not simplify.)

verify the identity.\n\\frac{\\sin(7x)-\\sin(x)}{\\cos(7x)-\\cos(x)}=-\\cot(4x)\n\nstart with the numerator of the left side and apply the appropriate formula of sum - to - product.\n\\sin(7x)-\\sin(x)=\\square (do not simplify.)

Answer

Explanation:

Step1: Recall sum - to - product formula

The sum - to - product formula for $\sin A-\sin B$ is $2\cos\left(\frac{A + B}{2}\right)\sin\left(\frac{A - B}{2}\right)$. Here $A = 7x$ and $B=x$. So, $\sin(7x)-\sin(x)=2\cos\left(\frac{7x + x}{2}\right)\sin\left(\frac{7x - x}{2}\right)$.

Step2: Simplify the expressions inside cosine and sine

Calculate $\frac{7x + x}{2}$ and $\frac{7x - x}{2}$. $\frac{7x+x}{2}=\frac{8x}{2}=4x$ and $\frac{7x - x}{2}=\frac{6x}{2}=3x$. So, $\sin(7x)-\sin(x)=2\cos(4x)\sin(3x)$.

Answer:

$2\cos(4x)\sin(3x)$