Consider the triangle OPQ inscribed in the unit circle. The coordinates of point P is (cos t, sin t), Q (cos s, sin s). The angle POQ has measure s t.

Figure 1

Reorienting the triangle such that the side OP coincides with the x-axis, we obtain this figure (figure 2)

Applying the distance formula in figure 1, we get

Next, applying again the distance formula in figure 2,

From these equations, we conclude that

How do we then determine sin(s+t) using this fact?

Note that the angles (s+t) and -(s+t) are complementary angles (if you add them up, the result is or 90 degrees). If they are complementary, the sine of one of them equals the cosine of the other and vice-versa. Thus,

Regrouping the terms in the right-hand side of the equation,

Applying the formula cos(s - t) = cos s cos t + sin s sin t, we have

Note that s and s are complementary angles (if you add them up, you get or 90 degrees). So, the sine of the of one of them equals the cosine of the other and vice-versa. Thus, cos( s) becomes sin s and sin( s) becomes cos s.