Maclaurin expansion for product of 2 sine functions

[YehiaMedhat]

Expand
$$sin(5x)sin(2x)$$ for 3 terms
When I tried to get the coefficients for each term I have to deriviate too long functions
For instance the first derivative was: $$2sin(5x)cos(2x)+5sin(2x)cos(5x)$$So don't you have any ideas how could I compress those products and terms into fewer-term function?

 

[blamocur]

My first choice would be to convert this product to a sum of trigonometric functions. You can either work out the formulae yourself or find them on internet.
My second choice would be to expand each of the multiplicands separately, then multiply the results.

 

[YehiaMedhat]

What is the name of a rule that makes it easier?
In other words, what sould I type in the search bar?

 

[YehiaMedhat]

And thanks for the help

 

[Dr.Peterson]

What is the name of a rule that makes it easier?
In other words, what sould I type in the search bar?

Try "product-to-sum identities".

 

[Steven G]

My first choice would be to convert this product to a sum of trigonometric functions. You can either work out the formulae yourself or find them on internet.
My second choice would be to expand each of the multiplicands separately, then multiply the results.

What is the name of a rule that makes it easier?
In other words, what should I type in the search bar?

product to a sum of trigonometric functions

 

[blamocur]

product to a sum of trigonometric functions

I did not actually know what they are called

 

[YehiaMedhat]

Thank you ?

 

[Steven G]

product to a sum of trigonometric functions

I did not actually know what they are called

You still did a good job with naming it!