# Transformation of a Product of Trigonometric Functions into a Sum or Difference

1. Let us find by expanding them by known formulae

So the above formula reduces to

(Sum) â†” (Product)

Thus a sum can be changed to a product or a product can be changed to a sum or difference. With the same procedure, we get.

• Working Rule for changing product â†” sum or difference
Example 1:
Express 2sin 4Î¸ cos 2Î¸ as a sum or difference

Solution:
(1) Formula used is either (1) or (2) since both have sin and cos
(2) 4Î¸ > 2Î¸

Note:- If the angle for the cosine function is bigger, use the formula (2).

Example 2:
Express as a sum or difference

Solution:
Note: There is no formulae connecting cos and sin in this series.
Either change cos â†’ sin (or) sin â†’ cos

Sum reduces to sin 55Â° + sin 72Â° (cos 35Â° = sin 55Â°)
Now use formula (1)

The sum could be done by writing it as cos 35Â° + cos18Â° also.
The two answers may look different; but they are equivalent

Example 3:
Prove that

are supplementary angles

Try these:
1. Prove that
2. Prove that

Example 4:
If A + B + C = Ï€ , prove that sin 2A + sin 2B + sin 2C = 4 sin A sin B sin C