angle sum identity (original) (raw)

It is desired to prove the identities

and

Consider the figure

where we have

• ∘
  △⁢A⁢a⁢d≡△⁢C⁢c⁢b
• ∘
  △⁢B⁢b⁢a≡△⁢D⁢d⁢c
• ∘
  a⁢d=d⁢c=1.

Also, everything is Euclidean, and in particular, the interior angles of any triangle sum to π.

Call ∠⁢A⁢a⁢d=θ and ∠⁢b⁢a⁢B=ϕ. From the triangle , we have ∠⁢A⁢d⁢a=π2-θ and ∠⁢D⁢d⁢c=π2-ϕ, while the degenerate angle ∠⁢A⁢d⁢D=π, so that

We have, therefore, that the area of the pink parallelogram is sin⁡(θ+ϕ). On the other hand, we can rearrange things thus:

In this figure we see an equal pink area, but it is composed of two pieces, of areas sin⁡ϕ⁢cos⁡θ and cos⁡ϕ⁢sin⁡θ. Adding, we have

which gives us the first. From definitions, it then also follows that sin⁡(θ+π/2)=cos⁡(θ), and sin⁡(θ+π)=-sin⁡(θ). Writing