The “sum and difference” formulas often come in handy, but it’s not immediately obvious that they would be true.

\begin{align}

\sin(\alpha \pm \beta) &= \sin \alpha \cos \beta \pm \cos \alpha \sin \beta \\

\cos(\alpha \pm \beta) &= \cos \alpha \cos \beta \mp \sin \alpha \sin \beta

\end{align}So what I want to know is,

- How can I prove that these formulas are correct?
- More importantly, how can I understand these formulas intuitively?
Ideally, I’m looking for answers that make no reference to Calculus, or to Euler’s formula, although such answers are still encouraged, for completeness.

**Answer**

Here are my favorite diagrams:

As given, the diagrams put certain restrictions on the angles involved: neither angle, nor their sum, can be larger than 90 degrees; and neither angle, nor their difference, can be negative. The diagrams can be adjusted, however, to push beyond these limits. (See, for instance, this answer.)

Here’s a bonus mnemonic cheer (which probably isn’t as exciting to read as to hear):

Sine, Cosine,

Sign, Cosine, Sine!

Cosine, Cosine,Co-Sign, Sine, Sine!

The first line encapsulates the sine formulas; the second, cosine. Just drop the angles in (in order $\alpha$, $\beta$, $\alpha$, $\beta$ in each line), and know that “Sign” means to use the *same sign* as in the compound argument (“+” for angle sum, “-” for angle difference), while “Co-Sign” means to use the *opposite sign*.

**Attribution***Source : Link , Question Author : Tyler , Answer Author : Blue*