Question

Which shows all the exact solutions of 2sec²x-tan⁴x=-1 ? Give your answer in radians.

A. π /3 +kπ and 2π /3 +kπ
B. π /3 +2kπ and 5π /3 +2kπ
C. π /4 +2kπ , 3π /4 +2kπ , 5π /4 +2kπ , and 7π /4 +2kπ
D. π /3 +kπ , 2π /3 +kπ , 4π /3 +kπ , and 5π /3 +kπ

Answer 1

The exact solutions of the equation 2sec²x - tan⁴x = -1 are x = π/2 + kπ, where k is an integer.

Step-by-step explanation:

To find the exact solutions of the equation 2sec²x - tan⁴x = -1, we can start by rewriting sec²x and tan⁴x in terms of cosine and sine.

1. sec²x = 1/cos²x

2. tan²x = sin²x/cos²x

Using these substitutions, the equation becomes:

2(1/cos²x) - (sin²x/cos²x)² = -1

Simplifying further:

2cos²x - sin⁴x = -cos²x

3cos²x - sin⁴x = 0

Factoring out cos²x:

cos²x(3 - sin²x) = 0

From this equation, we can identify two possibilities for the solutions:

1. cos²x = 0, which occurs when x = π/2 + kπ, where k is an integer

2. 3 - sin²x = 0, which occurs when sin²x = 3, but since sin²x is always between 0 and 1, this equation has no real solutions.

Therefore, the exact solutions of the equation 2sec²x - tan⁴x = -1 are x = π/2 + kπ, where k is an integer.