Trigonometric Identities

Overview

Trigonometric identities are equations that involve trigonometric functions and are true for all values of the variables where both sides are defined. In this section, we will explore the fundamental trigonometric identities, such as the Pythagorean, reciprocal, and quotient identities, and how to use them to simplify expressions and solve equations.

Key Concepts

  • Pythagorean Identities:
    • \( \sin^2(x) + \cos^2(x) = 1 \)
    • \( 1 + \tan^2(x) = \sec^2(x) \)
    • \( 1 + \cot^2(x) = \csc^2(x) \)
  • Reciprocal Identities:
    • \( \csc(x) = \frac{1}{\sin(x)} \)
    • \( \sec(x) = \frac{1}{\cos(x)} \)
    • \( \cot(x) = \frac{1}{\tan(x)} \)
  • Quotient Identities:
    • \( \tan(x) = \frac{\sin(x)}{\cos(x)} \)
    • \( \cot(x) = \frac{\cos(x)}{\sin(x)} \)

Using Identities

Trigonometric identities are powerful tools for simplifying expressions and solving trigonometric equations. By recognizing and applying these identities, you can rewrite expressions in simpler or more convenient forms. Some common uses include:

  • Simplifying expressions involving trigonometric functions.
  • Solving trigonometric equations.
  • Transforming complex expressions into known forms for easier computation.

Practice Problems

  1. Simplify the expression \( \frac{1 - \cos^2(x)}{\sin(x)} \).
    Solution

    Use the Pythagorean identity \( \sin^2(x) + \cos^2(x) = 1 \). Thus, \( 1 - \cos^2(x) = \sin^2(x) \), so the expression simplifies to \( \frac{\sin^2(x)}{\sin(x)} = \sin(x) \).

  2. Simplify the expression \( \frac{\tan(x)}{1 + \tan^2(x)} \).
    Solution

    Use the Pythagorean identity \( 1 + \tan^2(x) = \sec^2(x) \). Thus, the expression simplifies to \( \frac{\tan(x)}{\sec^2(x)} = \sin(x) \cdot \cos(x) \).

  3. Simplify the expression \( \frac{1}{\sec(x)} + \frac{1}{\csc(x)} \).
    Solution

    Use the reciprocal identities \( \sec(x) = \frac{1}{\cos(x)} \) and \( \csc(x) = \frac{1}{\sin(x)} \). The expression becomes \( \cos(x) + \sin(x) \).

  4. Prove that \( \sin^2(x) = 1 - \cos^2(x) \).
    Solution

    Use the Pythagorean identity \( \sin^2(x) + \cos^2(x) = 1 \). Subtract \( \cos^2(x) \) from both sides to get \( \sin^2(x) = 1 - \cos^2(x) \).

  5. Simplify the expression \( \frac{\sin(x)}{1 - \cos(x)} \).
    Solution

    Multiply the numerator and denominator by \( 1 + \cos(x) \) to rationalize the denominator, which results in \( \frac{\sin(x)(1 + \cos(x))}{(1 - \cos^2(x))} \). Using the identity \( 1 - \cos^2(x) = \sin^2(x) \), the expression simplifies to \( \frac{1 + \cos(x)}{\sin(x)} \).

  6. Simplify the expression \( \frac{1 - \cos(x)}{\sin(x)} \).
    Solution

    Use the reciprocal identity \( \csc(x) = \frac{1}{\sin(x)} \) and recognize that the expression simplifies to \( \csc(x) - \cot(x) \).

  7. Solve the equation \( \sin(x) = \cos(x) \).
    Solution

    Divide both sides of the equation by \( \cos(x) \) to obtain \( \tan(x) = 1 \). Thus, \( x = \frac{\pi}{4} + n\pi \), where \( n \) is any integer.

  8. Solve the equation \( \tan(x) = 1 \).
    Solution

    The general solution is \( x = \frac{\pi}{4} + n\pi \), where \( n \) is any integer.

  9. Solve the equation \( 1 + \tan^2(x) = 2 \).
    Solution

    Subtract 1 from both sides to get \( \tan^2(x) = 1 \). Taking the square root of both sides gives \( \tan(x) = \pm 1 \), so \( x = \frac{\pi}{4} + n\pi \) or \( x = -\frac{\pi}{4} + n\pi \), where \( n \) is any integer.

  10. Simplify the expression \( \frac{1 + \cot^2(x)}{\csc^2(x)} \).
    Solution

    Use the identity \( \csc^2(x) = 1 + \cot^2(x) \). The expression simplifies to \( \frac{\csc^2(x)}{\csc^2(x)} = 1 \).

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