# Trigonometric Identities Solver

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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.

## Trigonometric 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:

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

## Trigonometric Identities Problems

Below you could see some examples

### Solved Examples

Question 1: Evaluate $sin 105^{\circ} sin 15^{\circ}$
Solution:
Using the identity formula,

$sin A sin B =$ $\frac{1}{2}$$(cos(A - B) - cos(A + B)) sin 105^{\circ} sin 15^{\circ} =$$\frac{1}{2}$$(cos(105^{\circ} - 15^{\circ}) - cos(105^{\circ} + 15^{\circ})) = \frac{1}{2}$$(cos(90^{\circ}) - cos(120^{\circ}))$

= $\frac{1}{2}$$(0 - (-$$\frac{1}{2}$$)) = \frac{1}{4} = 0.25 sin 105^{\circ} sin 15^{\circ} = 0.25 Question 2: Evaluate sin 105^{\circ} - sin 15^{\circ} Solution: Using the identity formula, sin A - sin B = 2cos ($$\frac{A + B}{2}$$)sin($$\frac{A - B}{2}$$) sin 105^{\circ} - sin 15^{\circ} = 2cos($$\frac{105^{\circ} + 15^{\circ}}{2}$$)sin($$\frac{105^{\circ} - 15^{\circ}}{2}$$) = 2 cos 60^{\circ} sin 45^{\circ} = 2($$\frac{1}{2}$$)($$\frac{\sqrt{2}}{2}$$)$

= $\frac{\sqrt{2}}{2}$

= $0.707$

$sin 105^{\circ} - sin 15^{\circ} = 0.707$