Sine angle addition identity - problem with proof

[syncmaster913n]

Hello,

I am trying to prove to myself, for fun/learning, the sine angle addition identity. Disregarding whether my approach is correct or not, I'm getting a strange algebraic result near the end that I am finding hard to explain. So I came up with this drawing:

And I want to prove that:

I did the following:

I wanted to get rid of the C from the denominator of the first term, so I divided the numerator and denominator by C:

Keeping in mind that (from the triangles image above):

What I wanted to do now is prove that:

The only way for this equation to hold however would be for this to be true:

And for this in turn to be true, a/c must equal 1, which clearly cannot be the case.

I'm guessing the mistake is trivial and right under my nose. Would someone be kind enough to point it out?

 

[HallsofIvy]

If I am reading this correctly, you are starting with sin(x+ y)= sin(x)cos(y)+ cos(x)sin(y) which is, after all, what you want to prove

In the very first step you appear to replace sin(x) with \(\displaystyle \frac{d}{e}\). Where did you get that? That triangle is not a right triangle!

 

[syncmaster913n]

Thank you very much - this was exactly my error - somehow I just "assumed" (since the one and the large one are) and just went with that assumption. Thanks again!