Formulae for Converting Product into Sum or Difference

How to remember the formulae for converting product into sum or difference?

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

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

2 cos X cos Y = cos (X + Y) + cos (X - Y) ………. (iii)

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

The following points will help us to remember the above four formulas:

(i)The product to be converted to sum or difference and should contain 2 as a factor.

(ii) The angles in sines or cosines of sum appear as ‘sum’ (i.e., X + Y) of the given angles X and Y.

(iii) The angles in sines or cosines of difference appear as ‘difference’ (i.e., X - Y) of the given angles X and Y.

(iv) In case of formula (i), we shall have the sum of two sines when the product consists of a pair of sine and cosine. The angle in sine (i.e. X) of product is greater than the angle of cosine (i.e. Y).

(v) In case of formula (ii), we shall have the difference of two sines when the product consists of a pair of cosine and sine. The angle in cosine (i.e. X) of product is greater than the angle of sine (i.e. Y).

(vi) In case of formula (iii), we shall have the sum of two cosines when the product consists of two cosines.

(v) In case of formula (iv), we shall have the difference of two cosines when the product consists of two sines.

(vi) In case of formula (i), (ii) and (iii) when the product consists of a pair of sine and cosine or two cosines we first write the sum (i.e. X + Y) and then the difference (i.e. X - Y) of the angles in the converted formula; but in case of formula

(iv) when the product consists of two sines we first write the difference and then the sum of the angle in the converted formula.

The following verbal statements will help us to remember the above four formulas:

For formula (i): 2 sin X cos Y = sin (sum) + sin (difference) (X > Y)

For formula (ii): 2 cos X sin Y = sin (sum) - sin (difference) (X > Y)

For formula (iii): 2 cos X cos Y = cos (sum) + cos (difference)

For formula (iv): 2 sin X sin Y = cos (difference) - cos (sum)

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Converting Product into Sum/Difference and Vice Versa