Trigonometry
Sum to product and product to sum formulas or identities
Sum to product formulas for the sine and the cosine functions
The product to sum formulas for the sine and cosine functions
Trigonometric identities, examples
Sum to product and product to sum formulas or identities
Sum to product formulas for the sine and the cosine functions
Adding the sum and difference formulas for the sine function,
sin (a + b) = sin a · cos b + cosa · sin b   (1)
sin (a - b) = sin a · cos b - cosa · sin b   (2)
 yields sin (a + b) + sin (a - b) = 2sin a · cos b
and by subtracting the second from the first identity,
sin (a + b) - sin (a - b) = 2cosa · sin b.
Then, substitute   a + b = x    and   a - b = y .
 By adding and subtracting these equalities we get
 thus, and
Using the same procedure for the cosine function,
cos (a + b) = cos a · cos b - sin a · sin b   (1)
cos (a - b) = cos a · cos b + sin a · sin b   (2)
by adding  (1) + (2) we get,         cos (a + b) + cos (a - b) = 2cos a · cos b
and subtracting   (1) - (2)           cos (a + b) - cos (a - b) = -2sin a · sin b
 substitute,  a + b = x  and  a - b = y   so that,
 thus, and
The product to sum formulas for the sine and cosine functions
By adding and subtracting addition formulas derived are following product to sum formulas,
 and
 and
Trigonometric identities, examples
Example:   Using known values, sin 60° = Ö3/2 and sin 45° = Ö2/2 evaluate sin 105°.
Solution:  Applying the sum formula for the sine function,  sin (a + b) = sin a · cos b + cosa · sin b
therefore,   sin 105° = sin (60° + 45°) = sin 60° · cos 45° + cos60° · sin 45°
 Example:   Prove the identity
 Solution:   Using the addition formula
 Example:   Verify the identity
Solution:   We divide the numerator and denominator on the left side by sin a and to the right side we use the cotangent formula for the difference of two angles, thus
Example:   Express sin 3x in terms of sin x.
Solution:   Using the sum formula and the double angle formula for the sine function,
sin 3x = sin (2x + x) = sin 2x · cos x + cos 2x · sin x = 2sin x cos x · cos x + (cos2 x - sin2 x) · sin x
= 2sin x · (1 - sin2 x) + (1 - 2sin2 x) · sin x = 3sin x - 4sin3 x.
 Example:   Prove the identity
 Solution:
Example:   Express the given difference  sin 61° - sin 59°  as a product.
 Solution:   Since
Example:   Prove the identity  sin a + sin (a + 120°) + sin (a + 240°) = 0.
Solution:   Applying the sum formula to the last two terms on the left side of the identity we get,
 Example:   Prove that
Solution:  Replace sin a by cos (p/2 - a) and cos a by sin (p/2 - a) and use the sum to product formula
Pre-calculus contents F