Trigonometry

\cos (α + β) = \frac{O E}{O C}

O E = O D - E D

E D = H B

\begin{aligned} \cos (α) & = \frac{O D}{O B} \\ O B \cos (α) & = O D \end{aligned}

\begin{aligned} \cos (β) & = \frac{O B}{O C} \\ O B & = O C \cos (β) \end{aligned}

\begin{aligned} O C \cos (β) \cos (α) & = O D \end{aligned}

\begin{aligned} \cos (α) & = \frac{H B}{C B} \\ C B \cos (α) & = E D \end{aligned}

\begin{aligned} \sin (β) & = \frac{C B}{O C} \\ C B & = O C \sin (β) \end{aligned}

O C \sin (β) \cos (α) = E D

\begin{aligned} \cos (α + β) & = \frac{O C \cos (β) \cos (α) - O C \sin (β) \cos (α)}{O C} \\ = \cos (α) \cos (β) - \sin (α) \cos (β) \end{aligned}

\sin (α + β) = \frac{C E}{O C}

C E = C H + H E

C H = C B \cos (α)

C B = O C \sin (β)

C H = O C \sin (β) \cos (α)

H E = B D = O B \sin (α)

O B = O C \cos (β)

H E = O C \cos (β) \sin (α)

\begin{aligned} \sin (α + β) & = \frac{O C \sin (β) \cos (α) + O C \cos (β) \sin (α)}{O C} \\ = \sin (α) \cos (β) + \cos (α) \sin (β) \end{aligned}