To solve the equation \( \tan^2{x} = 1 \) using reciprocal trigonometric functions, we can proceed as follows:

1. Recall that the tangent function can be expressed as:
   \[
   \tan{x} = \frac{\sin{x}}{\cos{x}}
   \]
   However, since we're using reciprocal trigonometric functions, let's use the fact that:
   \[
   \cot{x} = \frac{1}{\tan{x}}
   \]
   
2. Start by taking the square root of both sides of the equation:
   \[
   \tan^2{x} = 1 \implies \tan{x} = \pm 1
   \]

3. Now, using the reciprocal function for tangent:
   \[
   \cot{x} = \frac{1}{\tan{x}} = \frac{1}{\pm 1} = \pm 1
   \]
   So, we now have:
   \[
   \cot{x} = \pm 1
   \]

4. The cotangent function is equal to \( 1 \) at angles where \( \tan{x} = 1 \), which occurs at:
   \[
   x = \frac{\pi}{4} + n\pi, \text{ where } n \in \mathbb{Z}
   \]
   Similarly, \( \cot{x} = -1 \) at angles where \( \tan{x} = -1 \), which occurs at:
   \[
   x = \frac{3\pi}{4} + n\pi, \text{ where } n \in \mathbb{Z}
   \]

Thus, the general solutions to \( \tan^2{x} = 1 \) are:
\[
x = \frac{\pi}{4} + n\pi, \quad n \in \mathbb{Z}
\]

Would you like more details on any specific part of the solution?

Here are 5 related questions:
1. How do you solve trigonometric equations involving other identities like \( \sin^2{x} = 1 \)?
2. What are the periodic properties of the tangent function?
3. How can we solve \( \tan^2{x} = 1 \) using the unit circle?
4. What is the relationship between the tangent and cotangent functions in terms of their graphs?
5. Can we solve \( \tan^2{x} = 1 \) using only basic algebra?

**Tip**: Always remember that tangent and cotangent functions are periodic with a period of \( \pi \), so their solutions repeat at regular intervals.

Learn how to solve the equation tan^2(x) = 1 using reciprocal trigonometric functions like cotangent. This solution involves key trigonometric identities and covers the periodic properties of tangent and cotangent. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation