Need help please! Convert the sum to product and simplify

[greed409]

I can simplify these problems for 2 terms but I cant figure out 4 terms for the life of me. Here is the specific question:
Convert the sum to the product and simplify: sin3t + sin5t + sin7t + sin9t
I would be ever so greatful if someone could help me solve this. Thank you so much!

 

[royhaas]

Just apply the identity \(\displaystyle sin(x+y)+sin(x-y)=2sin(x)cos(y)\) repeatedly.

 

[soroban]

Re: Need help please! Convert the sum to product and simplif

Hello, greed409!

Convert the sum to a product and simplify: \(\displaystyle \,\sin3t\,+\,\sin5t\,+\, \sin7t\,+\,\sin9t\)

Use the Sum-to-Product Formulas:

\(\displaystyle \;\;\sin A\,+\,\sin B\;=\;2\cdot\sin\left(\frac{A+B}{2}\right)\cdot\cos\left(\frac{A-B}{2}\right)\)

\(\displaystyle \;\;\cos A\,+\,\cos B\;=\;2\cdot\cos\left(\frac{A+B}{2}\right)\cdot\cos\left(\frac{A-B}{2}\right)\)


I got: \(\displaystyle \,4\cdot\sin(6t)\cdot\cos(2t)\cdot\cos(t)\)
\(\displaystyle \;\;\)But what do they mean by "and simplify"?