Question

use the sum - to - product identities to rewrite the following expression in terms containing only first powers of tangent.
\\(\frac{\sin7x - \sin5x}{\cos7x + \cos5x}\\)

Explanation:

Step1: Apply sum - to - product identities

The sum - to - product identities are $\sin A-\sin B = 2\cos\frac{A + B}{2}\sin\frac{A - B}{2}$ and $\cos A+\cos B=2\cos\frac{A + B}{2}\cos\frac{A - B}{2}$.
For $\frac{\sin7x-\sin5x}{\cos7x+\cos5x}$, substituting $A = 7x$ and $B = 5x$, we get $\frac{2\cos\frac{7x + 5x}{2}\sin\frac{7x - 5x}{2}}{2\cos\frac{7x+5x}{2}\cos\frac{7x - 5x}{2}}$.

Step2: Simplify the expression

Cancel out the common factor $2\cos\frac{7x + 5x}{2}=2\cos6x$ (assuming $\cos6x
eq0$). The expression becomes $\frac{\sin\frac{7x - 5x}{2}}{\cos\frac{7x - 5x}{2}}$.
Since $\frac{7x - 5x}{2}=x$, the expression is $\frac{\sin x}{\cos x}$.

Step3: Recall the tangent identity

We know that $\tan x=\frac{\sin x}{\cos x}$.

Answer:

$\tan x$