We have to rewrite the expression using sum-to-product identities.
The expression is:
[tex]\frac{\sin7x+\sin5x}{\cos7x+\cos5x}[/tex]We will then use this identity:
[tex]\sin a+\sin b=2\sin (\frac{a+b}{2})\cos (\frac{a-b}{2})[/tex]and this identity:
[tex]\cos a+\cos b=2\cos (\frac{a+b}{2})\cos (\frac{a-b}{2})[/tex]We can apply it to our expression as:
[tex]\begin{gathered} \frac{\sin7x+\sin5x}{\cos7x+\cos5x}=\frac{2\cdot\sin (\frac{7x+5x}{2})\cos (\frac{7x-5x}{2})}{2\cdot\cos (\frac{7x+5x}{2})\cos (\frac{7x-5x}{2})} \\ \frac{\sin7x+\sin5x}{\cos7x+\cos5x}=\frac{\sin (\frac{12x}{2})}{\cos (\frac{12x}{2})} \\ \frac{\sin7x+\sin5x}{\cos7x+\cos5x}=\frac{\sin (6x)}{\cos (6x)} \\ \frac{\sin7x+\sin5x}{\cos7x+\cos5x}=\tan (6x) \end{gathered}[/tex]Answer: the expression is equal to tan(6x)