Trigonometric Functions of Sum

trigonometric functions of sum and difference of two angles:

The trigonometric functions of sum and difference of two angles A are B are listed below:
Sin (A+B) = Sin (A) Cos (B) + Cos (A) Sin (B) .................. (1)
Sin (A-B) = Sin (A) Cos (B) - Cos (A) Sin (B) .................... (2)
Cos (A+B) = Cos (A) Cos (B) - Sin (A) Sin (B) ................... (3)
Cos (A-B) = Cos (A) Cos (B) + Sin (A) Sin (B) ................... (4)
Adding (1) and (2) we get, 
Sin (A+B) + Sin (A-B) = 2 Sin (A) Cos (B)
Subtracting (1) and (2), we get,
Sin (A+B) - Sin (A-B) = 2 Cos (A) Sin (B)
Similarly, adding (3) and (4), we get,
Cos (A+B) + Cos (A-B) = 2 Cos (A) Cos (B)
Subtracting (3) and (4), we get,
Cos (A+B) - Cos (A-B) = - 2 sin (A) sin (B)
These can be written in the forms
Sin (A+B) + Sin (A-B) = 2 Sin (A) Cos (B) ............................ (5)
Sin (A+B) - Sin (A-B) = 2 Cos (A) Sin (B) ............................. (6)
Cos (A+B) + Cos (A-B) = 2 Cos (A) Cos (B) ........................ (7)
Cos (A - B) - Cos (A+B) = 2 sin (A) sin (B) ........................... (8)
Note: The order on the right-hand side of (8) must be carefully noted
The sum and difference of two angles A and B of a Tangent function are given below:
Tan (A+B) = [Tan (A) + Tan (B)] / [1 - Tan (A) Tan (B)] .................. (9)
Tan (A-B) = [Tan (A) - Tan (B)] / [1 + Tan (A) Tan (B)] ................... (10)

Example for trigonometric functions of sum and difference of two angles

 Express as the sum of two trigonometrical ratios 
sin (5θ) cos (3θ)
Sol : Using the Identity:  2 Sin (A) Cos (B) = Sin (A+B) + Sin (A-B)  
On substitution, we get,
sin (5θ) cos (3θ) = (1/2) {Sin (5θ+3θ) + Sin (5θ-3θ)  
= (1/2) {Sin (8θ) + Sin (2θ) 

More examples for trigonometric functions of sum and difference of two angles

Change in to sum of trigonometric functions: Sin (70^o) Sin (20^o)
Sol: Using,   2 sin (A) sin (B) = Cos (A - B) - Cos (A+B)
Sin (70^o) Sin (20^o) = (1/2) { Cos (70^o - 20^o) - Cos (70^o+20^o) }
                            = (1/2) { Cos (50^o) - Cos (90^o) }
= (1/2) {Cos (50^o) }           Since Cos(90^o) = 0