These identities allow us to express a product of trigonometric functions as a sum or difference . They are derived from the addition and subtraction formulas.
2 sin A cos B = sin ( A + B ) + sin ( A − B )
2 cos A sin B = sin ( A + B ) − sin ( A − B )
2 cos A cos B = cos ( A − B ) + cos ( A + B )
2 sin A sin B = cos ( A − B ) − cos ( A + B )
Using the addition formulas:
sin ( A + B ) = sin A cos B + cos A sin B sin ( A − B ) = sin A cos B − cos A sin B
Adding: sin ( A + B ) + sin ( A − B ) = 2 sin A cos B
Subtracting: sin ( A + B ) − sin ( A − B ) = 2 cos A sin B
Similarly from cosine formulas:
cos ( A − B ) = cos A cos B + sin A sin B cos ( A + B ) = cos A cos B − sin A sin B
Adding: cos ( A − B ) + cos ( A + B ) = 2 cos A cos B
Subtracting: cos ( A − B ) − cos ( A + B ) = 2 sin A sin B
These identities convert a sum or difference of trigonometric functions into a product . Let A = 2 C + D B = 2 C − D
sin C + sin D = 2 sin ( 2 C + D ) cos ( 2 C − D )
sin C − sin D = 2 cos ( 2 C + D ) sin ( 2 C − D )
cos C + cos D = 2 cos ( 2 C + D ) cos ( 2 C − D )
cos C − cos D = − 2 sin ( 2 C + D ) sin ( 2 C − D )
Express 2 cos 75° sin 15°
Solution: Using 2 cos A sin B = sin ( A + B ) − sin ( A − B ) 2 cos 75° sin 15° = sin ( 75° + 15° ) − sin ( 75° − 15° ) = sin 90° − sin 60° = 1 − 2 3
Simplify cos 3 θ + cos θ sin 3 θ + sin θ
Solution:
Numerator: 2 sin ( 2 3 θ + θ ) cos ( 2 3 θ − θ ) = 2 sin 2 θ cos θ Denominator: 2 cos ( 2 3 θ + θ ) cos ( 2 3 θ − θ ) = 2 cos 2 θ cos θ ∴ 2 c o s 2 θ c o s θ 2 s i n 2 θ c o s θ = c o s 2 θ s i n 2 θ = tan 2 θ
Prove that cos 20° cos 40° cos 80° = 8 1
Solution: Multiply and divide by 2 sin 20° cos 20° cos 40° cos 80° = 2 s i n 20° 2 s i n 20° c o s 20° c o s 40° c o s 80° = 2 s i n 20° s i n 40° c o s 40° c o s 80° = 4 s i n 20° s i n 80° c o s 80° = 8 s i n 20° s i n 160° = 8 s i n 20° s i n ( 180° − 20° ) = 8 s i n 20° s i n 20° = 8 1
Product Form Sum/Difference Form 2 sin A cos B sin ( A + B ) + sin ( A − B ) 2 cos A sin B sin ( A + B ) − sin ( A − B ) 2 cos A cos B cos ( A − B ) + cos ( A + B ) 2 sin A sin B cos ( A − B ) − cos ( A + B )