Trigonometric Identity

The sum-to-product trigonometric identities

is a set of formulas that express the sum or difference of trigonometric functions in terms of products of trigonometric functions. For the sum of angles (a) and (b), the identity is:

\[\sin(a + b) = \sin a \cos b + \cos a \sin b \]

\[\cos(a + b) = \cos a \cos b - \sin a \sin b\]

\[\tan(a + b) = \frac{\tan a + \tan b}{1 - \tan a \tan b}\]

Derivation:

These identities are useful when you need to simplify expressions involving sums or differences of trigonometric functions.

Additionally, there are similar identities for cotangent, secant, and cosecant, and they can be derived from the sine and cosine identities.

The product-to-sum trigonometric identities

the product of trigonometric functions as a sum of trigonometric functions. There are two main product-to-sum identities, one for the sine function and another for the cosine function:

\[\sin(a)\sin(b) = \frac{1}{2}[\cos(a-b) - \cos(a+b)] \]

\[\cos(a)\cos(b) = \frac{1}{2}[\cos(a-b) + \cos(a+b)] \]