# Sum to Product Trigonometry Identities Calculator

Calculate the values for sin(u)+sin(v), sin(u)-sin(v), cos(u)+cos(v) and cos(u)-cos(v) against selected values of u and v angle.

## Using following formulas:

sin(u)+sin(v) = 2(sin(^{ u +}

_{2}

^{v}) ( cos(

^{ u -}

_{2}

^{v}))

sin(u)-sin(v) = 2(cos(

^{ u +}

_{2}

^{v}) ( sin(

^{ u -}

_{2}

^{v}))

cos(u)+cos(v) = 2(cos(

^{ u +}

_{2}

^{v}) ( cos(

^{ u -}

_{2}

^{v}))

cos(u)-cos(v) = -2(sin(

^{ u +}

_{2}

^{v}) ( sin(

^{ u -}

_{2}

^{v}))

The sum-to-product identities are the trigonometry statements that tells how to convert the summation or subtraction of 2-trigonometry functions into product of 2-trigonometry functions as shown in above formulas.

The sum-to-product identities deal only with sine and cosine functions.

The are the true trigonometry statements that tell you how to turn the sum or subtraction of two trig functions into the product of two trig functions.

### 👉 Proof Sum to Product Identities

### Applicaton and examples

1. The sum to product formulas are used to express the sum and difference of trigonometric functions sin and cos as products of sin and cos functions.

2. In the derivation of sum to product formula using the product to sum formulas in trigonometry.

3. We can apply these formulas to simplify trigonometric problems.

For example: Prove that (cos 4x - cos 2x) / (sin 4x + sin 2x) = - tan x

Solution: We will use the following sum to product formulas to prove:

cos A - cos B = -2 sin [(A + B)/2] sin [(A - B)/2]

sin A + sin B = 2 sin [(A + B)/2] cos [(A - B)/2]

now,

LHS = (cos 4x - cos 2x) / (sin 4x + sin 2x)

= -2 sin [(4x + 2x)/2] sin [(4x - 2x)/2] / 2 sin [(4x + 2x)/2] cos [(4x - 2x)/2]

= - [ sin (6x/2) sin (2x/2) ] / [ sin (6x/2) cos (2x/2)]

= - (sin 3x sin x) / (sin 3x cos x)

= - sin x / cos x

= - tan x

= RHS

Hence, we have proved (cos 4x - cos 2x) / (sin 4x + sin 2x) = - tan x using sum to product formulas.