If 2 Tan ⁡ X 1 − Tan ⁡ 2 X = 1 \frac{2 \tan X}{1-\tan ^2 X}=1 1 − T A N 2 X 2 T A N X ​ = 1 , Then X X X Can Equal:A. X = 7 Π 8 + N Π X=\frac{7 \pi}{8}+n \pi X = 8 7 Π ​ + Nπ B. X = 5 Π 8 + N Π X=\frac{5 \pi}{8}+n \pi X = 8 5 Π ​ + Nπ C. X = Π 8 + N Π X=\frac{\pi}{8}+n \pi X = 8 Π ​ + Nπ D. X = 3 Π 8 + N Π X=\frac{3 \pi}{8}+n \pi X = 8 3 Π ​ + Nπ

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Introduction

In this article, we will explore the given trigonometric equation $\frac{2 \tan x}{1-\tan ^2 x}=1$ and determine the possible values of $x$ that satisfy this equation. We will use various trigonometric identities and properties to simplify the equation and find the solutions.

Understanding the Equation

The given equation is $\frac{2 \tan x}{1-\tan ^2 x}=1$. To simplify this equation, we can use the identity $\tan ^2 x + 1 = \sec ^2 x$. However, in this case, we can use the identity $\tan ^2 x = \frac{\sin ^2 x}{\cos ^2 x}$ and $\sec ^2 x = \frac{1}{\cos ^2 x}$.

Simplifying the Equation

We can start by simplifying the equation using the identity $\tan ^2 x = \frac{\sin ^2 x}{\cos ^2 x}$. We have:

$$\frac{2 \tan x}{1-\tan ^2 x}=1$$

$$\frac{2 \tan x}{1-\frac{\sin ^2 x}{\cos ^2 x}}=1$$

$$\frac{2 \tan x}{\frac{\cos ^2 x - \sin ^2 x}{\cos ^2 x}}=1$$

$$\frac{2 \tan x \cos ^2 x}{\cos ^2 x - \sin ^2 x}=1$$

Using Trigonometric Identities

We can use the identity $\tan x = \frac{\sin x}{\cos x}$ to simplify the equation further. We have:

$$\frac{2 \tan x \cos ^2 x}{\cos ^2 x - \sin ^2 x}=1$$

$$\frac{2 \frac{\sin x}{\cos x} \cos ^2 x}{\cos ^2 x - \sin ^2 x}=1$$

$$\frac{2 \sin x \cos x}{\cos ^2 x - \sin ^2 x}=1$$

Simplifying Further

We can simplify the equation further by using the identity $\cos ^2 x - \sin ^2 x = \cos 2x$. We have:

$$\frac{2 \sin x \cos x}{\cos ^2 x - \sin ^2 x}=1$$

$$\frac{2 \sin x \cos x}{\cos 2x}=1$$

Using Double Angle Formula

We can use the double angle formula $\sin 2x = 2 \sin x \cos x$ to simplify the equation further. We have:

$$\frac{2 \sin x \cos x}{\cos 2x}=1$$

$$\frac{\sin 2x}{\cos 2x}=1$$

Solving for $x$

We can solve for $x$ by using the identity $\tan x = \frac{\sin x}{\cos x}$. We have:

$$\frac{\sin 2x}{\cos 2x}=1$$

$$\tan 2x = 1$$

Finding the Solutions

We can find the solutions for $x$ by using the identity $\tan 2x = 1$. We have:

$$\tan 2x = 1$$

$$2x = \frac{\pi}{4} + n \pi$$

$$x = \frac{\pi}{8} + \frac{n \pi}{2}$$

Checking the Solutions

We can check the solutions by plugging them back into the original equation. We have:

$$\frac{2 \tan x}{1-\tan ^2 x}=1$$

$$\frac{2 \tan (\frac{\pi}{8} + \frac{n \pi}{2})}{1-\tan ^2 (\frac{\pi}{8} + \frac{n \pi}{2})}=1$$

Conclusion

In this article, we have explored the given trigonometric equation $\frac{2 \tan x}{1-\tan ^2 x}=1$ and determined the possible values of $x$ that satisfy this equation. We have used various trigonometric identities and properties to simplify the equation and find the solutions. The solutions are given by $x = \frac{\pi}{8} + \frac{n \pi}{2}$.

Final Answer

The final answer is $\boxed{\frac{\pi}{8} + n \pi}$.

Discussion

The given equation is a trigonometric equation that involves the tangent function. We have used various trigonometric identities and properties to simplify the equation and find the solutions. The solutions are given by $x = \frac{\pi}{8} + \frac{n \pi}{2}$. This equation is a classic example of a trigonometric equation that involves the tangent function.

Related Topics

• Trigonometric identities
• Trigonometric equations
• Double angle formula
• Tangent function

References

• [1] "Trigonometry" by Michael Corral
• [2] "Trigonometric Equations" by Paul Dawkins
• [3] "Double Angle Formula" by Math Open Reference

Keywords

• Trigonometric equation
• Tangent function
• Double angle formula
• Trigonometric identities
• Trigonometric equations
  

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Q&A: Trigonometric Equation

Q: What is the given trigonometric equation?

A: The given trigonometric equation is $\frac{2 \tan x}{1-\tan ^2 x}=1$.

Q: What is the goal of the problem?

A: The goal of the problem is to find the possible values of $x$ that satisfy the given trigonometric equation.

Q: What trigonometric identities were used to simplify the equation?

A: The trigonometric identities used to simplify the equation are $\tan ^2 x = \frac{\sin ^2 x}{\cos ^2 x}$, $\sec ^2 x = \frac{1}{\cos ^2 x}$, and $\cos ^2 x - \sin ^2 x = \cos 2x$.

Q: What is the final solution for $x$?

A: The final solution for $x$ is $x = \frac{\pi}{8} + \frac{n \pi}{2}$.

Q: How can we check the solutions?

A: We can check the solutions by plugging them back into the original equation.

Q: What are some related topics to this problem?

A: Some related topics to this problem are trigonometric identities, trigonometric equations, double angle formula, and tangent function.

Q: What are some references for further reading?

A: Some references for further reading are "Trigonometry" by Michael Corral, "Trigonometric Equations" by Paul Dawkins, and "Double Angle Formula" by Math Open Reference.

Q: What are some keywords related to this problem?

A: Some keywords related to this problem are trigonometric equation, tangent function, double angle formula, trigonometric identities, and trigonometric equations.

Q&A: Solutions

Q: What are the possible values of $x$ that satisfy the equation?

A: The possible values of $x$ that satisfy the equation are $x = \frac{\pi}{8} + \frac{n \pi}{2}$.

Q: How can we find the solutions for $x$?

A: We can find the solutions for $x$ by using the identity $\tan 2x = 1$ and solving for $x$.

Q: What is the significance of the double angle formula in this problem?

A: The double angle formula is used to simplify the equation and find the solutions for $x$.

Q: How can we check the solutions for $x$?

A: We can check the solutions for $x$ by plugging them back into the original equation.

Q&A: Conclusion

Q: What is the final conclusion of the problem?

A: The final conclusion of the problem is that the possible values of $x$ that satisfy the equation are $x = \frac{\pi}{8} + \frac{n \pi}{2}$.

Q: What are some key takeaways from this problem?

A: Some key takeaways from this problem are the use of trigonometric identities to simplify the equation, the importance of the double angle formula, and the need to check the solutions.

Q: What are some future directions for this problem?

A: Some future directions for this problem are to explore other trigonometric equations and to investigate the properties of the tangent function.

Q&A: Final Thoughts

Q: What are some final thoughts on this problem?

A: Some final thoughts on this problem are that it is a classic example of a trigonometric equation that involves the tangent function, and that it requires the use of various trigonometric identities and properties to simplify the equation and find the solutions.

Q: What are some recommendations for further reading?

A: Some recommendations for further reading are "Trigonometry" by Michael Corral, "Trigonometric Equations" by Paul Dawkins, and "Double Angle Formula" by Math Open Reference.

Q: What are some keywords related to this problem?

A: Some keywords related to this problem are trigonometric equation, tangent function, double angle formula, trigonometric identities, and trigonometric equations.