Let \alpha and \beta be acute angles whose sum is also acute, such that $\angle BAC = \alpha,$ $\angle DAB = \beta,$ and $AB = 1$. Let $E$ be the foot of the perpendicular from $A$ to $BC$, and $F$ be the foot of the perpendicular from $E$ to $AB$, as shown below:

a) Show that triangle DEF is similar to triangle BCD.

b) Calculate $AC,$ $BC,$ $AD,$ $BD,$ $BF,$ and $DF$ in terms of trigonometric functions of $\alpha$ and $\beta$.

c) Use the diagram above to provide another proof of the sine and cosine angle sum identities for acute angles $\alpha$ and $\beta$ whose sum is below $90^{\circ}$.

siIviajendeukie Jun 11, 2024