If sinA+sinB=1/2 and cosA+cosB=1/2 then the value of tanA+tanB
-2/3
proof:
1. observe symmetry in A and B
2. Square each equation after sending terms of A/B on other side. We can prove:
SinA + CosA= 1/2 and SinB +CosB=1/2
3. bring equations in form of tan's by dividing CosA and then using SecA to TanA formulae.
4. Solve the quadratic obtained and the sum of the solutions is the sum of TanA and TanB
To find the value of tanA+tanB, we first need to recall the trigonometric identities for tangent. The tangent of an angle A is equal to the sine of A divided by the cosine of A.
So, let's start by rearranging the given equations involving sine and cosine to express sine and cosine in terms of tangent.
Given:
sinA + sinB = 1/2
cosA + cosB = 1/2
We know that sinA = tanA * cosA and sinB = tanB * cosB. By substituting these expressions into the first equation, we can solve for cosA and cosB:
tanA * cosA + tanB * cosB = 1/2 [Equation 1]
Similarly, we also know that cosA = 1/sqrt(1 + tan^2(A)) and cosB = 1/sqrt(1 + tan^2(B)). By substituting these expressions into the second equation, we obtain:
1/sqrt(1 + tan^2(A)) + 1/sqrt(1 + tan^2(B)) = 1/2 [Equation 2]
Now, we have a system of equations (Equations 1 and 2) involving tanA, tanB, cosA, and cosB. We can solve this system of equations to find the values of tanA and tanB.
Unfortunately, the system of equations is not straightforward to solve algebraically. However, we can use numerical methods, such as iteration or approximation techniques, to estimate the values of tanA and tanB that satisfy the given equations.
Therefore, without further information or constraints, we cannot determine the exact value of tanA+tanB.