the graph with the equation y = (sin 5x - sin 7x)/(cos 5x - cos 7x) is shown in a 0,2π, π/2 by -2,2,1…

the graph with the equation y = (sin 5x - sin 7x)/(cos 5x - cos 7x) is shown in a 0,2π, π/2 by -2,2,1 viewing rectangle. a. describe the graph using another equation. b. verify that the two equations are equal. a. write another equation of the given graph. y = - cot(6x) (type an equation using x as the variable.) b. to verify that the two equations are equal, start with the numerator of the right side of the first equation and apply the appropriate sum - to - product formula. sin 5x - sin 7x = (do not simplify.)

the graph with the equation y = (sin 5x - sin 7x)/(cos 5x - cos 7x) is shown in a 0,2π, π/2 by -2,2,1 viewing rectangle. a. describe the graph using another equation. b. verify that the two equations are equal. a. write another equation of the given graph. y = - cot(6x) (type an equation using x as the variable.) b. to verify that the two equations are equal, start with the numerator of the right side of the first equation and apply the appropriate sum - to - product formula. sin 5x - sin 7x = (do not simplify.)

Answer

Explanation:

Step1: Recall sum - to - product formula for sine

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 $A = 5x$ and $B=7x$. So, $\sin5x-\sin7x=2\cos\left(\frac{5x + 7x}{2}\right)\sin\left(\frac{5x - 7x}{2}\right)=2\cos(6x)\sin(-x)$. Since $\sin(-x)=-\sin x$, we have $\sin5x-\sin7x=- 2\cos(6x)\sin(x)$.

Step2: Recall sum - to - product formula for cosine

The sum - to - product formula for $\cos A-\cos B$ is $-2\sin\left(\frac{A + B}{2}\right)\sin\left(\frac{A - B}{2}\right)$. Here $A = 5x$ and $B = 7x$. So, $\cos5x-\cos7x=-2\sin\left(\frac{5x + 7x}{2}\right)\sin\left(\frac{5x - 7x}{2}\right)=-2\sin(6x)\sin(-x)=2\sin(6x)\sin(x)$.

Step3: Simplify the original fraction

We have $y=\frac{\sin5x-\sin7x}{\cos5x - \cos7x}=\frac{-2\cos(6x)\sin(x)}{2\sin(6x)\sin(x)}$. Canceling out the non - zero $\sin(x)$ (for $x$ such that $\sin(x)\neq0$), we get $y =-\cot(6x)$.

Answer:

$2\cos(6x)\sin(-x)$