Sin A  Sin B
Sin A  Sin B is an important trigonometric identity in trigonometry. It is used to find the difference of values of sine function for angles A and B. It is one of the difference to product formulas used to represent the difference of sine function for angles A and B into their product form. The result for Sin A  Sin B is given as 2 cos ½ (A + B) sin ½ (A  B).
Let us understand the Sin A  Sin B formula and its proof in detail using solved examples.
1.  What is Sin A  Sin B Identity in Trigonometry? 
2.  Sin A  Sin B Difference to Product Formula 
3.  Proof of Sin A  Sin B Formula 
4.  How to Apply Sin A  Sin B? 
5.  FAQs on Sin A  Sin B 
What is Sin A  Sin B Identity in Trigonometry?
The trigonometric identity Sin A  Sin B is used to represent the difference of sine of angles A and B, Sin A  Sin B in the product form with the help of the compound angles (A + B) and (A  B). Let us study the Sin A  Sin B formula in detail in the following sections.
Sin A  Sin B Difference to Product Formula
The Sin A  Sin B difference to product formula in trigonometry for angles A and B is given as,
Sin A  Sin B = 2 cos ½ (A + B) sin ½ (A  B)
Here, A and B are angles, and (A + B) and (A  B) are their compound angles.
Proof of Sin A  Sin B Formula
We can give the proof of Sin A  Sin B formula using the expansion of sin(A + B) and sin(A  B) formula. As we stated in the previous section, we write Sin A  Sin B = 2 cos ½ (A + B) sin ½ (A  B).
Let us assume two compound angles A and B, given as A = X + Y and B = X  Y,
⇒ Solving, we get,
X = (A + B)/2 and Y = (A  B)/2
We know, sin(X + Y) = sin X cos Y + sin Y cos X
sin(X  Y) = sin X cos Y  sin Y cos X
sin(X + Y)  sin(X  Y) = 2 sin Y cos X
⇒ sin A  sin B = 2 sin ½ (A  B) cos ½ (A + B)
⇒ sin A  sin B = 2 cos ½ (A + B) sin ½ (A  B)
Hence, proved.
How to Apply Sin A  Sin B?
Sin A  Sin B trigonometric formula can be applied as a difference to the product identity to make the calculations easier when it is difficult to calculate the sine of the given angles. Let us understand its application using an example of sin 60º  sin 30º. We will solve the value of the given expression by 2 methods, using the formula and by directly applying the values, and compare the results. Have a look at the belowgiven steps.
 Compare the angles A and B with the given expression, sin 60º  sin 30º. Here, A = 60º, B = 30º.
 Solving using the expansion of the formula Sin A  Sin B, given as, Sin A  Sin B = 2 cos ½ (A + B) sin ½ (A  B), we get,
Sin 60º  Sin 30º = 2 cos ½ (60º + 30º) sin ½ (60º  30º) = 2 cos 45º sin 15º = 2 (1/√2) ((√3  1)/2√2) = (√3  1)/2.  Also, we know that Sin 60º  Sin 30º = (√3/2  1/2) = (√3  1)/2.
Hence, the result is verified.
☛ Topics Related to Sin A  Sin B:
Examples Using Sin A  Sin B Identity

Example 1: Find the value of sin 145º  sin 35º using sin A + sin B identity.
Solution:
We know, Sin A  Sin B = 2 cos ½ (A + B) sin ½ (A  B)
Here, A = 145º, B = 35º
sin 145º  sin 35º = 2 cos ½ (145º + 35º) sin ½ (65º  35º)
= 2 cos 90º cos 15º
= 0 [∵ cos 90º = 0]

Example 2: Using the values of angles from the trigonometric table, solve the expression: 2 cos 67.5º sin 22.5º
Solution:
We can rewrite the given expression as, 2 cos 67.5º sin 22.5º = 2 cos ½ (135)º sin ½ (45)º
Assuming A + B = 135º, A  B = 45º and solving for A and B, we get, A = 90º and B = 45º.
⇒ 2 cos ½ (135)º sin ½ (45)º = 2 cos ½ (90º + 45º) sin ½ (90º  45º)
We know, Sin A  Sin B = 2 cos ½ (A + B) sin ½ (A  B)
2 cos ½ (90º + 45º) sin ½ (90º  45º) = sin 90º  sin 45º = 1  (1/√2).

Example 3: Solve the given expression, (sin x  sin 5x)/(sin x + sin 5x).
Solution:
We have,
(sin x  sin 5x)/(sin x + sin 5x) = [2 cos ½ (x + 5x) sin ½ (x  5x)]/[2 sin ½ (x + 5x) cos ½ (x  5x)]
= [cos 3x sin(2x)]/[sin 3x cos(2x)]
= cos 3x sin 2x/sin 3x cos 2x
=  tan 2x cot 3x

Example 4: Verify the given expression using expansion of Sin A  Sin B: sin 70º  cos 70º = √2 sin 25º
Solution:
We have, L.H.S. = sin 70º  cos 70º
Since, cos 70º = cos(90º  20º) = sin 20º
⇒ sin 70º  cos 70º = sin 70º  sin 20º
Using Sin A  Sin B = 2 cos ½ (A + B) sin ½ (A  B)
⇒ sin 70º  sin 20º = 2 cos ½ (70º + 20º) sin ½ (70º  20º)
= 2 cos 45º sin 25º
= √2 sin 25º
= R.H.S.
Hence, verified.
FAQs on Sin A  Sin B
What is Sin A  Sin B in Trigonometry?
Sin A  Sin B is an identity or trigonometric formula, used in representing the difference of sine of angles A and B, Sin A  Sin B in the product form using the compound angles (A + B) and (A  B). Here, A and B are angles.
How to Use Sin A  Sin B Formula?
To use Sin A  Sin B formula in a given expression, compare the expansion, Sin A  Sin B = 2 cos ½ (A + B) sin ½ (A  B) with given expression and substitute the values of angles A and B.
What is the Formula of Sin A  Sin B?
Sin A  Sin B formula, for two angles A and B, can be given as, Sin A  Sin B = 2 cos ½ (A + B) sin ½ (A  B). Here, (A + B) and (A  B) are compound angles.
What is the Expansion of Sin A  Sin B in Trigonometry?
The expansion of Sin A  Sin B formula is given as, Sin A  Sin B = 2 cos ½ (A + B) sin ½ (A  B), where A and B are any given angles.
How to Prove the Expansion of Sin A  Sin B Formula?
The expansion of Sin A  Sin B, given as Sin A  Sin B = 2 cos ½ (A + B) sin ½ (A  B), can be proved using the 2 sin Y cos X product identity in trigonometry. Click here to check the detailed proof of the formula.
What is the Application of Sin A  Sin B Formula?
Sin A  Sin B formula can be applied to represent the difference of sine of angles A and B in the product form of sine of (A  B) and cosine of (A + B), using the formula, Sin A  Sin B = 2 cos ½ (A + B) sin ½ (A  B).
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