Final answer:
To find tan²a, we used the Pythagorean identity to express cos²a in terms of sin²a and then substituted into the given equation. Simplifying and using the identity for tan²a, we found that tan²a = 7.
Step-by-step explanation:
To find tan²a given that 5sin²a + 13cos²a = 6, let's use the Pythagorean identity which states that sin²a + cos²a = 1. Since we are given an equation involving sin²a and cos²a, we can manipulate this identity to express tan²a in terms of sin²a or cos²a.
First, let's express cos²a in terms of sin²a using the identity: cos²a = 1 - sin²a. Substituting this into the given equation, we get 5sin²a + 13(1 - sin²a) = 6. Simplifying that:
5sin²a + 13 - 13sin²a = 6
-8sin²a = -7
sin²a = 7/8.
Next, we will use the identity for tan²a, which is tan²a = sin²a/cos²a. Now, we can substitute cos²a = 1 - sin²a into this identity: tan²a = (7/8) / (1 - 7/8), which simplifies to tan²a = (7/8) / (1/8)
= 7.
Therefore, tan²a = 7.