Sin(A+B).Sin(A-B)

Let's simplify the expression sin(A + B) * sin(A - B) step by step.

Step 1: Expand the terms using trigonometric identities

We know that the sine of a sum and difference can be expanded as:

• sin(A + B) = sin A * cos B + cos A * sin B
• sin(A - B) = sin A * cos B - cos A * sin B

Substituting these into the left-hand side (L.H.S), we get:

L.H.S 

= sin(A + B) * sin(A - B)          

= (sin A * cos B + cos A * sin B) * (sin A * cos B - cos A * sin B)    

Step 2: Expand the product

Now, let's expand the product using the distributive property:

= sin A * cos B * sin A * cos B - sin A * cos B * cos A * sin B + cos A * sin B * sin A * cos B - cos A * sin B * cos A * sin B    

Upon simplifying this, we can group the terms as follows:

= sin²A * cos²B - cos²A * sin²B    

Step 3: Simplifying further using Pythagorean identity

Next, we'll use the Pythagorean identity, sin²X + cos²X = 1, to simplify the terms:

= sin²A * (1 - sin²B) - (1 - sin²A) * sin²B    

Now, expand both terms:

= sin²A - sin²A * sin²B - sin²B + sin²A * sin²B    

Step 4: Combine like terms

Notice that the terms - sin²A * sin²B and + sin²A * sin²B cancel each other out. This leaves us with:

= sin²A - sin²B    

Step 5: Conclusion

Thus, the left-hand side simplifies to:

L.H.S = sin²A - sin²B    

Since this matches the right-hand side (R.H.S), we conclude that:

L.H.S = R.H.S    

This proves the identity: sin(A + B) * sin(A - B) = sin²A - sin²B.

Final Answer:

sin a b identity simplifies to sin a b as shown above, confirming the equality.