Final answer:
To find the sum of all possible values of tanA, given 1 + sin²A = 3sinA cosA, we must solve the quadratic equation 1 = tanA(3 - tan²A).
Step-by-step explanation:
The student is asked: If 1 + sin²A = 3sinA cosA, then sum of all possible values of tanA is:______. This is a Mathematics problem related to trigonometric identities. Let's first rearrange the equation to express everything in terms of sine and cosine:
1 + sin²A = 3sinA cosA
Now, recall that tanA is the ratio of sinA to cosA (tanA = sinA/cosA). To find the sum of all possible values of tanA, we need to find the values of A for which the above equation holds true.
Rewrite the equation as:
1 = 3sinA cosA - sin²A
1 = sinA(3cosA - sinA)
Because 1 = cos²A + sin²A, rewrite the equation again as:
cos²A = sinA(3cosA - sinA)
Divide both sides by cos²A to get:
1 = tanA(3 - tan²A)
This quadratic equation can now be solved for tanA. Once we have the values of tanA, we can add them to find the sum of all possible values of tanA.