If (cosA)/(cosB)=x/y, then (x tanA+y tanB)/(x+y)=?
To find the value of (x tanA + y tanB)/(x + y), we need to manipulate the given equation and use trigonometric identities.
Starting with the given equation:
(cosA)/(cosB) = x/y
Let's rewrite this equation using trigonometric identities. The identity we will use is:
tanθ = sinθ/cosθ
For cosine:
cosA = sinA/tanA [Multiplying both sides by tanA]
cosB = sinB/tanB [Multiplying both sides by tanB]
Substituting these identities back into the given equation, we get:
(sinA/tanA)/(sinB/tanB) = x/y
Rearranging the equation, we have:
(x tanA)/(y tanB) = (sinA/sinB)
Now, let's find the value of (x tanA + y tanB)/(x + y) using the given equation. We can rewrite (x tanA + y tanB)/(x + y) as:
[(x/y) * (tanA)] + [(1) * (tanB)] / [(x/y) + (1)]
Substituting the given equation (sinA/sinB) = (x tanA)/(y tanB), we get:
[(sinA/sinB) * (tanA)] + [(1) * (tanB)] / [(sinA/sinB) + (1)]
Simplifying this expression further, we have:
[(sinA * tanA) + (sinB * tanB)] / [(sinA + sinB)]
Therefore, the value of (x tanA + y tanB)/(x + y) is:
[(sinA * tanA) + (sinB * tanB)] / [(sinA + sinB)]