Sum and Difference Identities

The sum identity for sine states that $$\sin(a + b) = \sin a \cos b + \cos a \sin b$$ .

The difference identity for sine states that $$\sin(a - b) = \sin a \cos b - \cos a \sin b$$ .

The sum identity for cosine states that $$\cos(a + b) = \cos a \cos b - \sin a \sin b$$ .

The difference identity for cosine states that $$\cos(a - b) = \cos a \cos b + \sin a \sin b$$ .

$$\sin(75^{o}) = \sin(33^{o} + 42^{o}) = \sin 33^{o} \cos 42^{o} + \cos 33^{o} \sin 42^{o}$$ .

$$\cos(\frac{\pi}{5} + \frac{\pi}{6}) = \cos\frac{\pi}{5} \cos\frac{\pi}{6} - \sin\frac{\pi}{5} \sin\frac{\pi}{6}$$ .

The value is $$\cos(20 + 40) = \cos(60) = \frac{1}{2}$$ .

The value is $$\cos(75^{o} + 15^{o}) = \cos(90^{o}) = 0$$ .

The value is $$\sin(5\pi/12 - \pi/4) = \sin(\pi/3) = \frac{\sqrt{3}}{2}$$ .

Sine and cosine are related through their identities, where sine can be expressed in terms of cosine and vice versa, especially in sum and difference formulas.