If 1/sinA+1/cosA =1/sinB+1/cosB, prove that: cot(A/2+B/2) = TanA. TanB
To prove the given statement, we need to show that cot(A/2 + B/2) is equal to tanA * tanB.
1. Start by simplifying the expression 1/sinA + 1/cosA = 1/sinB + 1/cosB.
To do this, we need to find a common denominator for the fractions. We can do this by multiplying the numerator and denominator of the first fraction by cosA, and the numerator and denominator of the second fraction by sinA.
After multiplying, the equation becomes:
(cosA + sinA) / (sinA * cosA) = (sinB + cosB) / (sinB * cosB)
2. Now, let's express the left-hand side (LHS) and the right-hand side (RHS) in terms of tangents and cotangents using trigonometric identities.
For the LHS, we can use the identity: tan(x) = sin(x) / cos(x)
Applying this identity to the LHS expression, we have:
(cosA + sinA) / (sinA * cosA) = tan(A) + cot(A)
For the RHS, we will follow a similar approach:
(sinB + cosB) / (sinB * cosB) = tan(B) + cot(B)
3. Now, substitute tan(A) + cot(A) for the LHS and tan(B) + cot(B) for the RHS in the given equation:
tan(A) + cot(A) = tan(B) + cot(B)
4. Rearrange the equation by subtracting cot(A) and tan(B) from both sides to isolate cot(A) - cot(B) = tan(B) - tan(A).
5. Apply the cotangent difference formula on the left-hand side and the tangent difference formula on the right-hand side.
cot(A) - cot(B) = -1 / tan(A + B) = -1 / tan(A + B)
tan(B) - tan(A) = -tan(A + B) / [1 + tan(A) * tan(B)]
6. Since cot(A) - cot(B) = -1 / tan(A + B) = tan(B) - tan(A) after simplification, we can say that:
-1 / tan(A + B) = -tan(A + B) / [1 + tan(A) * tan(B)]
7. Multiply both sides of the equation by [1 + tan(A) * tan(B)] to eliminate the denominators:
-1 = -tan(A + B) * [1 + tan(A) * tan(B)]
8. Divide both sides of the equation by -1:
1 = tan(A + B) * [1 + tan(A) * tan(B)]
9. Divide both sides of the equation by [1 + tan(A) * tan(B)]:
tan(A + B) = 1 / [1 + tan(A) * tan(B)]
10. Notice that the right-hand side of the equation represents tan(A/2 + B/2) using the tangent sum formula.
Therefore, we have cot(A/2 + B/2) = tan(A) * tan(B).
Hence, we have proved that cot(A/2 + B/2) = tan(A) * tan(B) from the given equation 1/sinA + 1/cosA = 1/sinB + 1/cosB.