Compound angle identities are trigonometric formulas that express the sine, cosine, or tangent of the sum or difference of two angles in terms of the trigonometric functions of the individual angles.
The sine compound angle identity is:
$$\sin(\alpha \pm \beta) = \sin\alpha\cos\beta \pm \cos\alpha\sin\beta$$
Let's calculate $\sin(75°)$ using the compound angle identity:
We can write 75° as 45° + 30°. Then:
$\sin(75°) = \sin(45° + 30°) = \sin(45°)\cos(30°) + \cos(45°)\sin(30°)$
$= \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2} = \frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4} = \frac{\sqrt{6} + \sqrt{2}}{4}$
The cosine compound angle identity is:
$$\cos(\alpha\pm\beta) = \cos\alpha\cos\beta \mp \sin\alpha\sin\beta$$
A sneaky part of the cosine compound angle formula is the $\mp$ sign on the right-hand side. This means that if you have $\cos(\alpha + \beta)$, you have to flip the sign to turn it into $\cos\alpha\cos\beta - \sin\alpha\sin\beta$, and vice versa for $\cos(\alpha - \beta)$.
We can derive the compound tangent identity from the compound sine and cosine identities.
First, we know that:
$$\tan(\alpha \pm \beta) = \frac{\sin(\alpha \pm \beta)}{\cos(\alpha \pm \beta)}$$
We can therefore substitute the identities for compound sine and cosine:
$$\tan(\alpha \pm \beta) = \frac{\sin\alpha\cos\beta \pm \cos\alpha\sin\beta}{\cos\alpha\cos\beta \mp \sin\alpha\sin\beta}$$
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