Question

There exist constants a, b, c, and d such that(sin x)^7 = asin 7x + bsin 5x + csin 3x + dsin x for all angles x. Find d.

I tried rewriting them in the exponential form, but it becomes ugly as there are many negative cosines and divisions of i. Help?

Answer 1

http://www.wolframalpha.com/input/?i=%28sinx%29%5E7

will show the answer you need.

sin^7 = sin(1-cos^2)^3
Now you can start expanding and using the sum formulas.

I'm sure a google search will show up all the gory details.

Answer 2

Or, you can go the other way

sin(7x) = sin(6x)cos(x) + cos(6x)sin(x)

and start expanding those as well, till all you have left is a bunch of sin(kx).