Question

Use the sum-to-product identities to rewrite the following expression in terms containing only firstpowers of tangent.sin7x + sin5xcos 7x + cos5x

Answer 1

We have to rewrite the expression using sum-to-product identities.

The expression is:

`(\sin7x+\sin5x)/(\cos7x+\cos5x)`

We will then use this identity:

`\sin a+\sin b=2\sin ((a+b)/(2))\cos ((a-b)/(2))`

and this identity:

`\cos a+\cos b=2\cos ((a+b)/(2))\cos ((a-b)/(2))`

We can apply it to our expression as:

`\begin{gathered} (\sin7x+\sin5x)/(\cos7x+\cos5x)=(2\cdot\sin ((7x+5x)/(2))\cos ((7x-5x)/(2)))/(2\cdot\cos ((7x+5x)/(2))\cos ((7x-5x)/(2))) \\ (\sin7x+\sin5x)/(\cos7x+\cos5x)=(\sin ((12x)/(2)))/(\cos ((12x)/(2))) \\ (\sin7x+\sin5x)/(\cos7x+\cos5x)=(\sin (6x))/(\cos (6x)) \\ (\sin7x+\sin5x)/(\cos7x+\cos5x)=\tan (6x) \end{gathered}`

Answer: the expression is equal to tan(6x)