We will learn how to deal with the formula for converting sum or difference into product.

(i) the sum of two sines into a product of a pair of sine and cosine

(ii) the difference of two sines into a product of a pair of cosine and sine

(iii) the sum of two cosines into a product of two cosines

(iv) the difference of two cosines into a product of two sines

**If X and Y are any two real numbers or angles, then**

(a) sin (X + Y) + sin (X - Y) = 2 sin X cos Y

(b) sin (X + Y) - sin (X - Y) = 2 cos X sin Y

(c) cos (X + Y) + cos (X - Y) = 2 cos X cos Y

(d) cos (X - Y) - cos (X + Y) = 2 sin X sin Y

(a), (b), (c) and (d) are considered as formulae of transformation from sum or difference to product.

**Proof:**

(a) We know that sin (X + Y) = sin X cos Y + cos X sin Y ……… (i)

and sin (X - Y) = sin X cos Y - cos X sin Y ……… (ii)

Adding (i) and (ii) we get,

**sin (X + Y) + sin (X
- Y) = 2 sin X cos Y **………………..… (1)

(b) We know that sin (X + Y) = sin X cos Y + cos X sin Y ……… (i)

and sin (X - Y) = sin X cos Y - cos X sin Y ……… (ii)

Subtracting (ii) from (i) we get,

**sin (X + Y) - sin (X
- Y) = 2 cos X sin Y **………………..… (2)

(c) We know that cos (X + Y) = cos X cos Y + sin X sin Y ……… (iii)

and cos (X - Y) = cos X cos Y - sin X sin Y ……… (iv)

Adding (iii) and (iv) we get,

**cos (X + Y) + cos (X
- Y) = 2 cos X cos Y **………………..… (3) ** **

(d) We know that cos (X + Y) = cos X cos Y + sin X sin Y ……… (iii)

and cos (X - Y) = cos X cos Y - sin X sin Y ……… (iv)

Subtracting (iii) from (iv) we get,

**cos (X - Y) - cos (X
+ Y) = 2 sin X sin Y **………………..… (4)

Let, X + Y = α and X - Y = β.

Then, we have, X = (α + β)/2 and B = (α - β)/2.

Clearly, formula (1), (2), (3) and (4) reduce to the following forms in terms of C and D:

sin α + sin β = 2 sin (α + β)/2 cos (α - β)/2 ………. (5)

sin α - sin β = 2 cos (α + β)/2 sin (α - β)/2 ……… (6)

cos α + cos β = 2 cos (α + β)/2 cos (α - β)/2 ……… (7)

And cos α - cos β = -2 sin (α + β)/2 sin (α - β)/2

⇒ cos α - cos β = 2 sin (α + β)/2 sin (β - α)/2 ……… (8)

**Note:** (i) Formula sin α + sin β = 2 sin (α + β)/2 cos (α - β)/2
is transform the sum of two sines into a product of a pair of sine and cosine.

(ii) Formula sin α - sin β = 2 cos (α + β)/2 sin (α - β)/2 is transform the difference of two sines into a product of a pair of cosine and sine.

(iii) Formula cos α + cos β = 2 cos (α + β)/2 cos (α - β)/2 is transform the sum of two cosines into a product of two cosines.

(iv) Formula cos α - cos β = 2 sin (α + β)/2 sin (β - α)/2 is transforms the difference of two cosines into a product of two sines.

**●**** Converting Product into Sum/Difference and Vice Versa**

**Converting Product into Sum or Difference****Formulae for Converting Product into Sum or Difference****Converting Sum or Difference into Product****Formulae for Converting Sum or Difference into Product****Express the Sum or Difference as a Product****Express the Product as a Sum or Difference**

**11 and 12 Grade Math**

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