SinA + sinB/sinA - sinB= tanA+B/2cotA-B/2
Check your sum-to-product formulas. I expect it will fall right out.
To simplify the given expression, we can start by working with one side at a time. Let's begin with the left side of the equation:
(sinA + sinB) / (sinA - sinB)
To simplify this expression, we can use the identity:
a^2 - b^2 = (a + b)(a - b)
Applying this identity, we can rewrite the numerator as:
(sinA + sinB) = sinA + sinB
And the denominator as:
(sinA - sinB) = sinA - sinB
Now we can simplify further:
sinA + sinB / sinA - sinB = (sinA + sinB) / (sinA - sinB)
Next, let's simplify the right side of the equation:
tan(A + B) / 2cot(A - B) / 2
Using the definitions of tangent (tan) and cotangent (cot), we can rewrite this expression:
tan(A + B) = (sin(A + B)) / (cos(A + B))
cot(A - B) = (cos(A - B)) / (sin(A - B))
Now let's substitute these values back into the equation:
(sin(A + B)) / (cos(A + B)) / 2((cos(A - B)) / (sin(A - B)))
To simplify further, we can multiply the numerator and denominator by (sin(A - B)):
(sin(A + B))(sin(A - B)) / (cos(A + B))(sin(A - B)) / 2(cos(A - B))
Using the product-to-sum identities, we can simplify the numerator:
sin(A + B)(sin(A - B)) = (sinA * cosB + cosA * sinB)(sinA * cosB - cosA * sinB)
Now, let's simplify the denominator:
(cos(A + B))(sin(A - B)) = (cosA * cosB - sinA * sinB)(sinA * cosB - cosA * sinB)
Now, let's substitute these simplified expressions back into the equation:
[(sinA * cosB + cosA * sinB)(sinA * cosB - cosA * sinB)] / [(cosA * cosB - sinA * sinB)(sinA * cosB - cosA * sinB)] / 2(cos(A - B))
Simplifying further by canceling out the common terms and simplifying the remaining expression, we get:
1 / 2
Therefore, the simplified form of the given expression is 1 / 2.