We will learn how to deal with the formula for converting product into sum or difference.
(i) the product of a pair of sine and cosine into the sum of two sines
(ii) the product of a pair of cosine and sine into the difference of two sines
(iii) the product of two cosines into the sum of two cosines
(iv) the product of two sines into the difference of two cosines
If X and Y are any two real numbers or angles, then
(a) 2 sin X cos Y = sin (X + Y) + sin (X  Y)
(b) 2 cos X sin Y = sin (X + Y)  sin (X  Y)
(c) 2 cos X cos Y = cos (X + Y) + cos (X  Y)
(d) 2 sin X sin Y = cos (X  Y)  cos (X + Y)
(a), (b), (c) and (d) are considered as formulae of transformation from product to sum or difference.
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,
2 sin X cos Y = sin (X + Y) + sin (X  Y)
(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,
2 cos X sin Y = sin (X + Y)  sin (X  Y)
(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,
2 cos X cos Y = cos (X + Y) + cos (X  Y)
(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,
2 sin X sin Y = cos (X  Y)  cos (X + Y)
11 and 12 Grade Math
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