Pre calc cos(15x)+cos(6x)

Question

Write the sum as a product.

cos(15x) + cos(6x)

Answer 1

Housney's work is correct. It really boils down into what you consider to be a trig. identity.

If we consider there to be levels of identity, at level 1, we see

cos(a + b) = cos a cos b - sin a sin b

at level 2, we derive

cos a cos b = 1/2 (cos(a-b) + cos(a+b)) , Housney's identity, from this first formula; it also uses the cos difference formula, but the difference formula could also be viewed as a next level formula as it is so easily derived from the sum formula and knowledge that cos -a = cos b and sin -a = - sin a

at level 3, we have cos a + cos b = 2 cos ((a+b)/2))((a-b)/2)

This identity is "kind of" derived by Housney.

You might as well just start with this formula.

Answer 2

cos A cos B = (1/2)​ (cos(A−B) + cos(A+B))

we need to find two numbers A and B such that:

A + B = 15 .................(1)

A - B = 6 .................(2)

Adding (1) and (2), we obtain:

2A = 21 ------------> A = 21/2

B = 15 - A = 15 - 21/2 = 9/2 -----------------> B = 9/2

cos(15x) + cos(6x) = 2 cos(21x / 2) cos(9x / 2)