sin A cos B+cos A sin B/cos A cos B-sin A sin B? =tan A +tan B/1-tan A tan B
Now, we will multiply the numerator and the denominator on the left-hand side by 1cos A cos B. Note that this does not change the value of the expression because a fraction with the same numerator and denominator is equal to 1.
sin A cos B+cos A sin B/cos A cos B-sin A sin B * 1cos A cos B/1cos A cos B
⇕
(sin A cos B+cos A sin B) * 1cos A cos B/(cos A cos B-sin A sin B) * 1cos A cos B
We will simplify the numerator and the denominator of the above expression one at a time. Let's start with the numerator!
We have simplified the denominator.
tan A + tan B/1-tan A tan B
We started with the left-hand side of the given identity, modified it using other known relationships, and arrived at the right-hand side.
tan (A+B)? =tan A +tan B/1-tan A tan B
Simplify LHS
tan A + tan B/1-tan A tan B = tan A +tan B/1-tan A tan B ✓
