Angles add, but sines do not: sin 75° is not sin 45° + sin 30°. A picture with two stacked right triangles proves sin(a + b) = sin a cos b + cos a sin b and cos(a + b) = cos a cos b - sin a sin b. Dividing gives tan(a + b), and the even and odd rules turn each into a subtraction formula. With them the crew finds exact values for 15°, 75° and 105°, and setting a = b gives the double-angle formulas.