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
11 and 12 Grade Math
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