Proof of Sum to Product Formulas
Proof of Sum to Product formulas are added below,
Proof of sin A + sin B = 2 sin[(A+B)/2] cos[(A-B)/2]
From the product to sum formula we can write:
2 sin P cos Q = sin (P + Q) + sin (P β Q)
Putting P = [(A+B)/2] and Q = [(A-B)/2]
Substituting these values in above equation we get,
2 sin[(A+B)/2] cos [(A-B)/2] = sin[{(A+B)/2} + {(A-B)/2}] + sin[{(A+B)/2} β {(A-B)/2}]
2 sin[(A+B)/2] cos [(A-B)/2] = sin A + sin B
sin A + sin B = 2 sin[(A+B)/2] cos[(A-B)/2]
Hence Proved
Proof of sin A β sin B = 2 cos[(A+B)/2] sin[(A-B)/2]
From the product to sum formula we can write:
2 cos P sin Q = sin (P + Q) β sin (P β Q)
Putting P = [(A+B)/2] and Q = [(A-B)/2]
Substituting these values in above equation we get,
2 cos[(A+B)/2] sin [(A-B)/2] = sin[{(A+B)/2} + {(A-B)/2}] β sin[{(A+B)/2} β {(A-B)/2}]
2 cos[(A+B)/2] sin [(A-B)/2] = sin A β sin B
sin A β sin B = 2 cos[(A+B)/2] sin[(A-B)/2]
Hence Proved
Proof of cos A + cos B = 2 cos[(A+B)/2] cos[(A-B)/2]
From the product to sum formula we can write:
2 cos P cos Q = cos (P + Q) + cos (P β Q)
Putting P = [(A+B)/2] and Q = [(A-B)/2]
Substituting these values in above equation we get,
2 cos[(A+B)/2] cos [(A-B)/2] = cos[{(A+B)/2} + {(A-B)/2}] + cos[{(A+B)/2} β {(A-B)/2}]
2 cos[(A+B)/2] cos [(A-B)/2] = cos A+ cos B
cos A + cos B = 2 cos[(A+B)/2] cos[(A-B)/2]
Hence Proved
Proof of cos A β cos B = 2 sin[(A+B)/2] sin[(A-B)/2]
From the product to sum formula we can write:
2 sin P sin Q = cos (P + Q) β cos (P β Q)
Putting P = [(A+B)/2] and Q = [(A-B)/2]
Substituting these values in above equation we get,
2 sin[(A+B)/2] sin [(A-B)/2] = cos[{(A+B)/2} + {(A-B)/2}] β cos[{(A+B)/2} β {(A-B)/2}]
2 sin[(A+B)/2] sin [(A-B)/2] = cos A β cos B
cos A β cos B = 2 sin[(A+B)/2] sin[(A-B)/2]
Hence Proved
Sum to Product Formulas
The sum to product formulas are trigonometric identities that convert the sum or difference of two trigonometric functions into a product of trigonometric functions. These formulas are particularly useful in simplifying expressions, solving trigonometric equations, and integrating functions.
Sum to Product formulas are important formulas of trigonometry. Four sum-to-product formulas in trigonometry are,
- sin A + sin B = 2 sin [(A+B)/2] Γ cos [(A-B)/2]
- sin A β sin B = 2 cos[(A+B)/2] Γ sin[(A-B)/2]
- cos A + cos B = 2 cos[(A+B)/2] Γ cos[(A-B)/2]
- cos A β cos B = 2 sin[(A+B)/2] Γ sin[(A-B)/2]
In this article, we will learn about Sum to Product Formulas, Proof of Sum to Product Formulas, Application of Sum to Product Formulas in detail.
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