Trigonometric Identity is a study of identities that involve trigonometric functions. An "identity" is an equation of one or more variables that holds true for all values. These identities involves some functions of one or more angles.

             There are many fundamental identities like Reciprocal Identities, Ratio Identities, Opposite angle Identities, Pythagorean Identities, Cofunction Identities, Sum and Difference Identities, Double angle Identities, Product to Sum Identities, Sum to Product Identities etc.

        This calculator is only concentrating on product to sum identities and sum to product identities.
Below you can see identities                                      Product to Sum Identities:

$$sin A sin B = \frac{1}{2}(cos(A - B) - cos(A + B))$$
$$cos A cos B = \frac{1}{2}(cos(A - B) + cos(A + B))$$
$$sin A cos B = \frac{1}{2}(sin(A + B) + sin(A - B))$$   
$$cos A sin B = \frac{1}{2}(sin(A + B) - sin(A - B))$$   
                                               Sum to Product Identities:

$$sin A + sin B = 2sin\left ( \frac{A + B}{2} \right )cos\left ( \frac{A - B}{2} \right )$$
$$sin A - sin B = 2cos\left ( \frac{A + B}{2} \right )sin\left ( \frac{A - B}{2} \right )$$
$$cos A + cos B = 2cos\left ( \frac{A + B}{2} \right )cos\left ( \frac{A - B}{2} \right )$$
$$cos A - cos B = 2sin\left ( \frac{A + B}{2} \right )sin\left ( \frac{A - B}{2} \right )$$