Prove: sin(A+B)+sin(A-B)/sin(A+B)-sin(A-B)=tanAcotB
To prove the identity sin(A+B) + sin(A-B) / sin(A+B) - sin(A-B) = tanA cotB, we will work on simplifying the left-hand side of the equation and then equate it with the right-hand side.
1. Start with the left-hand side:
sin(A+B) + sin(A-B) / sin(A+B) - sin(A-B)
2. Apply the common denominator to simplify the expression:
(sin(A+B) * (sin(A+B) - sin(A-B)) + sin(A-B) * (sin(A+B) + sin(A-B))) / (sin(A+B) - sin(A-B))
3. Expand and combine like terms in the numerator:
(sin(A+B)*sin(A+B) - sin(A+B)*sin(A-B) + sin(A-B)*sin(A+B) + sin(A-B)*sin(A-B)) / (sin(A+B) - sin(A-B))
4. Simplify the numerator further:
(sin(A+B)*sin(A+B) + sin(A-B)*sin(A+B) - sin(A+B)*sin(A-B) + sin(A-B)*sin(A-B)) / (sin(A+B) - sin(A-B))
5. Observe that the two middle terms in the numerator will cancel each other out:
(sin(A+B)*sin(A+B) + sin(A-B)*sin(A-B)) / (sin(A+B) - sin(A-B))
6. Recognize that sin^2(x) - sin^2(y) = sin^2(x) / sin^2(y) - 1, which is a trigonometric identity:
(sin(A+B) + sin(A-B)) / (sin(A+B) - sin(A-B))
7. Use the sum-to-product formula sin(x+y) + sin(x-y) = 2cos(y)sin(x) to rewrite the numerator:
(2cosBsinA) / (sin(A+B) - sin(A-B))
8. Apply the difference-to-product formula sin(x+y) - sin(x-y) = 2sin(x)cos(y) to rewrite the denominator:
(2cosBsinA) / (2sinAcosB)
9. Cancel out the common factors of 2, sinA, and cosB:
cotB / tanA
10. Finally, use the fact that cot(x) is equal to 1/tan(x):
1/tanB / tanA
cotB / tanA
Since the left-hand side is equal to the right-hand side, the original identity is proved.