Solving $2 \text{sec}^2 X - \text{tan}^4 X = -1$ Exact Solutions In Radians

#h1

Trigonometric equations can often seem daunting, but with a systematic approach and a solid understanding of trigonometric identities, finding their exact solutions becomes a manageable task. This article will walk you through the process of solving the equation $2 ext{sec}^2 x - \text{tan}^4 x = -1$, providing a detailed explanation of each step and highlighting key concepts along the way. Understanding how to solve trigonometric equations is a crucial skill in mathematics, with applications in various fields such as physics, engineering, and computer science. Let's embark on this mathematical journey together!

Understanding the Problem

#h2

The problem at hand is to find all the exact solutions of the trigonometric equation $2 \text{sec}^2 x - \text{tan}^4 x = -1$, expressed in radians. Before diving into the solution, let's break down the key components of the problem. Exact solutions imply that we are looking for solutions that can be expressed in terms of radicals and $\pi$, rather than decimal approximations. Radians are the standard unit of angular measure in mathematics, and it's essential to express our solutions in this unit. The equation itself involves two trigonometric functions: the secant ($\text{sec}$) and the tangent ($\text{tan}$). To solve this equation effectively, we'll need to leverage trigonometric identities to simplify the equation and express it in terms of a single trigonometric function. Trigonometric identities are fundamental tools in simplifying and solving trigonometric equations. They provide relationships between different trigonometric functions, allowing us to rewrite expressions in a more manageable form. The Pythagorean identity, in particular, will play a crucial role in our solution. This identity states that $\sin^2 x + \cos^2 x = 1$, and it can be manipulated to relate $\text{sec}^2 x$ and $\text{tan}^2 x$. Remember that $\text{sec} x$ is the reciprocal of $\cos x$, and $\tan x$ is the ratio of $\sin x$ to $\cos x$. These definitions are essential for understanding the relationships between these functions and for applying the appropriate identities. Before we jump into the algebraic manipulations, it's always a good idea to have a strategy in mind. Our strategy will involve using the Pythagorean identity to rewrite the equation in terms of a single trigonometric function, then solving the resulting equation. We'll also need to consider the periodic nature of trigonometric functions to ensure we find all possible solutions. The periodic nature of trigonometric functions means that they repeat their values over regular intervals. For example, the sine and cosine functions have a period of $2\pi$, while the tangent function has a period of $\pi$. This periodicity means that there are infinitely many solutions to a trigonometric equation, and we need to express these solutions in a general form that captures all possible values. Keep in mind that our goal is not just to find one or two solutions, but to find all solutions. This requires a careful consideration of the periodicity of the trigonometric functions involved and the use of general solution formulas. With this understanding of the problem and our strategy in place, we are ready to embark on the first step of the solution: simplifying the equation using trigonometric identities.

Simplifying the Trigonometric Equation

#h2

To solve the equation $2 \text{sec}^2 x - \text{tan}^4 x = -1$, the first step is to simplify it using trigonometric identities. Our goal is to express the equation in terms of a single trigonometric function, which will make it easier to solve. Recall the Pythagorean identity that relates $\text{sec}^2 x$ and $\tan^2 x$: $1 + \tan^2 x = \text{sec}^2 x$. This identity is a cornerstone in simplifying trigonometric expressions, allowing us to convert between secant and tangent functions. By substituting this identity into our original equation, we can eliminate the $\text{sec}^2 x$ term and express the equation solely in terms of $\tan x$. This substitution is a crucial step in simplifying the equation. Substituting $1 + \tan^2 x$ for $\text{sec}^2 x$ in the original equation, we get: $2(1 + \tan^2 x) - \tan^4 x = -1$ Now, we can expand and rearrange the equation to get a polynomial equation in terms of $\tan x$. Expanding the equation, we have: $2 + 2\tan^2 x - \tan^4 x = -1$ Rearranging the terms, we obtain: $\tan^4 x - 2\tan^2 x - 3 = 0$ This equation is a quadratic equation in disguise. If we let $y = \tan^2 x$, we can rewrite the equation as: $y^2 - 2y - 3 = 0$ This quadratic equation is much easier to solve than the original trigonometric equation. Solving quadratic equations is a fundamental skill in algebra, and we can use various methods, such as factoring, completing the square, or the quadratic formula, to find the solutions. In this case, the equation is easily factorable, which simplifies the solution process. The act of substituting $y = \tan^2 x$ is a common technique used to simplify equations involving trigonometric functions raised to even powers. It allows us to treat the trigonometric function as a single variable, making the equation more manageable. This technique is particularly useful when dealing with equations involving powers of sine, cosine, tangent, and their reciprocals. By recognizing the quadratic structure of the equation, we can apply our knowledge of algebra to find the solutions for $y$, which will then lead us to the solutions for $x$. This step-by-step simplification process is a key strategy in solving trigonometric equations. By breaking down the equation into smaller, more manageable parts, we can systematically work towards the solution. The substitution of $y = \tan^2 x$ transforms the trigonometric equation into a familiar algebraic form, making it easier to solve. With the equation now in a quadratic form, we can proceed to the next step: solving for $y$.

Solving the Quadratic Equation

#h2

We've successfully transformed the original trigonometric equation into a quadratic equation in terms of $y$, where $y = \text{tan}^2 x$. The quadratic equation is: $y^2 - 2y - 3 = 0$ Now, we need to solve this equation for $y$. As mentioned earlier, there are several methods for solving quadratic equations, including factoring, completing the square, and using the quadratic formula. In this case, the equation is easily factorable, which is often the quickest and most straightforward method. Factoring involves expressing the quadratic expression as a product of two linear factors. This method relies on recognizing the factors of the constant term that add up to the coefficient of the linear term. Factoring is a powerful technique for solving quadratic equations, but it's not always applicable. If the equation is not easily factorable, we can resort to other methods, such as completing the square or the quadratic formula. However, when factoring is possible, it often provides a more elegant and efficient solution. To factor the quadratic equation $y^2 - 2y - 3 = 0$, we look for two numbers that multiply to -3 and add to -2. These numbers are -3 and 1. Therefore, we can factor the equation as: $(y - 3)(y + 1) = 0$ Now, we can apply the zero-product property, which states that if the product of two factors is zero, then at least one of the factors must be zero. This property is fundamental in solving equations by factoring. Applying the zero-product property, we set each factor equal to zero: $y - 3 = 0 \text or } y + 1 = 0$ Solving these linear equations for $y$, we get $y = 3 \text{ or y = -1$ These are the solutions for $y$, but remember that $y = \text{tan}^2 x$. So, we need to substitute back to find the values of $x$. Substituting back is a crucial step in solving equations where we've made a substitution. It allows us to express the solutions in terms of the original variable. Without substituting back, we would only have solutions for the substituted variable, which would not answer the original problem. Now, we have two equations to solve: $\text{tan}^2 x = 3 \text{ and } \text{tan}^2 x = -1$ Let's analyze each equation separately. The equation $\text{tan}^2 x = 3$ will lead to valid solutions for $x$, as the square of a real number can be positive. However, the equation $\text{tan}^2 x = -1$ has no real solutions, since the square of a real number cannot be negative. This is an important observation that simplifies our task. We can disregard the equation $\text{tan}^2 x = -1$ and focus solely on solving $\text{tan}^2 x = 3$. This step demonstrates the importance of checking the validity of solutions. Not all solutions obtained through algebraic manipulation are necessarily valid solutions to the original problem. In this case, the equation $\text{tan}^2 x = -1$ arises from the algebraic solution process, but it does not correspond to any real solutions for $x$. This highlights the need for careful analysis and verification of solutions in mathematics. With the quadratic equation solved and the invalid solution discarded, we are left with the equation $\text{tan}^2 x = 3$. The next step is to solve this equation for $x$, which will involve taking the square root and considering the periodicity of the tangent function.

Finding Solutions for x

#h2

We've narrowed down our problem to solving the equation $\text{tan}^2 x = 3$. To find the values of $x$, we first take the square root of both sides of the equation. Remember to consider both the positive and negative square roots, as squaring either a positive or negative number will result in a positive value. Taking the square root of both sides, we get: $\texttan } x = \pm \sqrt{3}$ This gives us two separate equations to solve $\text{tan x = \sqrt3} \text{ and } \text{tan } x = -\sqrt{3}$ Now, we need to find the angles $x$ whose tangent is equal to $\sqrt{3}$ or $-\sqrt{3}$. Recall the unit circle and the special trigonometric values for common angles. The tangent function is defined as the ratio of the sine to the cosine, so we are looking for angles where this ratio is equal to $\sqrt{3}$ or $-\sqrt{3}$. Understanding the unit circle and the values of trigonometric functions at key angles is crucial for solving trigonometric equations. The unit circle provides a visual representation of the sine and cosine functions, and it allows us to quickly determine the values of these functions at common angles such as $0, \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3},$ and $\frac{\pi}{2}$. By knowing these values, we can often solve trigonometric equations without relying on a calculator. For $\text{tan } x = \sqrt{3}$, we know that the tangent function is positive in the first and third quadrants. The reference angle whose tangent is $\sqrt{3}$ is $\frac{\pi}{3}$. Therefore, the solutions in the interval $[0, 2\pi)$ are $x = \frac{\pi}{3}$ and $x = \frac{\pi}{3} + \pi = \frac{4\pi}{3}$. The reference angle is the acute angle formed between the terminal side of an angle and the x-axis. It's a useful tool for finding solutions to trigonometric equations in different quadrants. By finding the reference angle and considering the sign of the trigonometric function in each quadrant, we can determine all the solutions within a given interval. For $\text{tan } x = -\sqrt{3}$, the tangent function is negative in the second and fourth quadrants. The reference angle is still $\frac{\pi}{3}$. Therefore, the solutions in the interval $[0, 2\pi)$ are $x = \pi - \frac{\pi}{3} = \frac{2\pi}{3}$ and $x = 2\pi - \frac{\pi}{3} = \frac{5\pi}{3}$. However, we need to express all the solutions, considering the periodicity of the tangent function. The tangent function has a period of $\pi$, which means that its values repeat every $\pi$ radians. This periodicity is a key characteristic of the tangent function and must be taken into account when finding all the solutions to an equation involving tangent. To account for the periodicity, we add integer multiples of $\pi$ to our solutions. Therefore, the general solutions are $x = \frac{\pi{3} + k\pi \text{ and } x = \frac{2\pi}{3} + k\pi$ where $k$ is an integer. These general solutions represent all possible solutions to the equation $\text{tan}^2 x = 3$. The general solution is a concise way of expressing all possible solutions to a trigonometric equation. It incorporates the periodicity of the trigonometric function by adding integer multiples of the period to the principal solutions. By understanding and using general solutions, we can ensure that we have captured all the solutions to a trigonometric equation. With the general solutions found, we have successfully solved the equation $\text{tan}^2 x = 3$. The next step is to present the final answer in the required format.

Final Answer

#h2

We have successfully navigated the steps to find the exact solutions of the equation $2 \text{sec}^2 x - \text{tan}^4 x = -1$. Let's recap the process:

1. We used the Pythagorean identity $1 + \tan^2 x = \text{sec}^2 x$ to rewrite the equation in terms of $\tan x$.
2. We obtained a quadratic equation in $\tan^2 x$ and solved it by factoring.
3. We discarded the extraneous solution $\text{tan}^2 x = -1$ and focused on $\text{tan}^2 x = 3$.
4. We took the square root of both sides, considering both positive and negative roots.
5. We found the solutions for $x$ in the interval $[0, 2\pi)$ and then generalized them by adding integer multiples of $\pi$ to account for the periodicity of the tangent function.

The general solutions we found are: $x = \frac\pi}{3} + k\pi \text{ and } x = \frac{2\pi}{3} + k\pi$ where $k$ is an integer. These solutions represent all the exact solutions of the original equation in radians. Now, we need to compare our solutions with the given options and choose the correct one. The options provided are in the same general form, so we can directly match our solutions to the options. By comparing our solutions with the options, we can identify the correct answer. This final step ensures that we have answered the question accurately and completely. The ability to compare solutions and identify the correct answer is a crucial skill in mathematics. It demonstrates a thorough understanding of the problem-solving process and the ability to verify the results. In this case, comparing our solutions with the options allows us to confirm that we have found the correct general solutions to the trigonometric equation. Therefore, the exact solutions of the equation $2 \text{sec}^2 x - \text{tan}^4 x = -1$ are $\frac{\pi{3} + k\pi \text{ and } \frac{2\pi}{3} + k\pi$ where $k$ is an integer. This completes the solution to the problem. We have successfully found all the exact solutions of the given trigonometric equation, expressed in radians, and presented them in a clear and concise manner. The solution process involved a combination of trigonometric identities, algebraic manipulation, and an understanding of the periodicity of trigonometric functions. By mastering these skills, you can confidently tackle a wide range of trigonometric equations. In conclusion, solving trigonometric equations requires a systematic approach, a strong foundation in trigonometric identities, and careful attention to detail. By following the steps outlined in this article, you can develop the skills and confidence to solve even the most challenging trigonometric problems.

Choose the correct answer

#h2

Based on the steps above, the answer is (A).