Final answer:
To find tan²a, we need to use the trigonometric identities and the given equation. Start with the given equation: 5sin²a+13cos²a=6. Simplify and solve for cos²a. Then, use the identity tan²a=(1−cos²a)/cos²a to find tan²a.
Step-by-step explanation:
To find tan²a, we need to use the trigonometric identities and the given equation:
Start with the given equation: 5sin²a + 13cos²a = 6
Since sin²a = 1 - cos²a, we can substitute it in the equation:
5(1 - cos²a) + 13cos²a = 6
Simplify and solve for cos²a:
5 - 5cos²a + 13cos²a = 6
8cos²a - 5 = 6
8cos²a = 11
cos²a = 11/8
Now, we can use the identity tan²a = (1 - cos²a) / cos²a to find tan²a:
tan²a = (1 - 11/8) / 11/8
tan²a = (8 - 11) / 11
tan²a = -3/11
Therefore, tan²a = -3/11.