Which shows all the exact solutions of $2 \sec ^2 x-\tan ^4 x=-1$ ? Give your answer in radians.

A. $\frac{\pi}{3}+k \pi$ and $\frac{2 \pi}{3}+k \pi$

B. $\frac{\pi}{3}+2 k \pi$ and $\frac{5 \pi}{3}+2 k \pi$

C. $\frac{\pi}{4}+2 k \pi, \frac{3 \pi}{4}+2 k \pi, \frac{5 \pi}{4}+2 k \pi$, and $\frac{7 \pi}{4}+2 k \pi$

D. $\frac{\pi}{3}+k \pi, \frac{2 \pi}{3}+k \pi, \frac{4 \pi}{3}+k \pi$, and $\frac{5 \pi}{3}+k \pi$

Rewrite the equation using the identity sec 2 x = 1 + tan 2 x .
Simplify the equation to a quadratic form by substituting u = tan 2 x .
Solve for u and then for tan x .
Find the general solutions for x : 3 π ​ + kπ and 3 2 π ​ + kπ ​

Explanation

Problem Analysis We are given the equation 2 sec 2 x − tan 4 x = − 1 and we need to find all exact solutions in radians.

Using Trigonometric Identity We can use the trigonometric identity sec 2 x = 1 + tan 2 x to rewrite the equation in terms of tan x . Substituting this into the given equation, we get:


2 ( 1 + tan 2 x ) − tan 4 x = − 1
2 + 2 tan 2 x − tan 4 x = − 1

Rewriting the Equation Rearranging the terms, we obtain a quartic equation in tan x :

tan 4 x − 2 tan 2 x − 3 = 0
Let u = tan 2 x . Then the equation becomes:
u 2 − 2 u − 3 = 0

Solving for u We can factor the quadratic equation in u as follows:

( u − 3 ) ( u + 1 ) = 0
So, u = 3 or u = − 1 .

Solving for tan(x) Since u = tan 2 x , we have tan 2 x = 3 or tan 2 x = − 1 . However, tan 2 x cannot be negative for real x , so we only consider tan 2 x = 3 . This gives us:

tan x = ± 3 ​

Finding the values of x Now we need to find the values of x for which tan x = 3 ​ and tan x = − 3 ​ .

For tan x = 3 ​ , the solutions are x = 3 π ​ + kπ , where k is an integer.
For tan x = − 3 ​ , the solutions are x = 3 2 π ​ + kπ , where k is an integer.

General Solutions Therefore, the general solutions for x are x = 3 π ​ + kπ and x = 3 2 π ​ + kπ , where k is an integer.

Final Answer The exact solutions of the equation 2 sec 2 x − tan 4 x = − 1 are x = 3 π ​ + kπ and x = 3 2 π ​ + kπ , where k is an integer.


Examples
Trigonometric equations like this one are useful in physics for modeling oscillations and wave phenomena. For example, the angle of a pendulum or the amplitude of an electromagnetic wave can be described using trigonometric functions. Solving these equations allows us to predict the behavior of these systems at different points in time or space. Understanding trigonometric equations is also crucial in fields like engineering, where they are used in the design of structures and circuits.

Answered by GinnyAnswer | 2025-07-03

The solutions for the equation 2 sec 2 x − tan 4 x = − 1 are x = 3 π ​ + kπ and x = 3 2 π ​ + kπ for integers k . These apply to the trigonometric functions where tan x = ± 3 ​ . Thus, the answer is Option A.
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Answered by Anonymous | 2025-07-04