Problem 26 - Algebra and Trigonometry

Question

Verify each identity. $$ (\sin \theta-\cos \theta)^{2}=1-\sin 2 \theta $$

Step-by-Step Solution

Answer

The identity \( (\sin \theta-\cos \theta)^{2}=1-\sin 2 \theta \) can be verified as correct by squaring the left side, applying the Pythagorean identity, and utilizing the double angle identity for sine.

Step 1: Square the Left Side

Squaring \( \sin \theta - \cos \theta \), we get \( \sin^2 \theta - 2 \sin \theta \cos \theta + \cos^2 \theta \).

Step 2: Use the Pythagorean Identity

Recognize that \( \sin^2 \theta + \cos^2 \theta = 1 \). So, our equation transforms to \(1 - 2 \sin \theta \cos \theta\).

Step 3: Utilize the Double Angle Identity

Recognize that \( \sin 2 \theta = 2 \sin \theta \cos \theta \). So, our equation transforms to \(1 - \sin 2 \theta\).

Key Concepts

Pythagorean Identity

The Pythagorean identity is one of the fundamental trigonometric identities in trigonometry. It establishes a relationship between the square of sine and cosine functions. To recall, the Pythagorean identity is expressed as:

• \( \sin^2 \theta + \cos^2 \theta = 1 \)

This identity is derived from the Pythagorean theorem and applies to all angles \( \theta \). It helps connect the sin and cos functions, demonstrating that despite their differences, they always sum up to unity when squared.
This identity is frequently used to simplify expressions or solve trigonometric equations. For instance, in the original exercise, squaring \( (\sin \theta - \cos \theta) \) led to discovering \( \sin^2 \theta + \cos^2 \theta = 1 \), which simplifies the equation significantly.
Recognizing and using the Pythagorean identity allows you to transform and solve complex problems or verify identities effectively. It's fundamental to understanding more advanced trigonometric concepts.

Double Angle Identity

Another key trigonometric identity is the double angle identity. This identity involves the trigonometric functions of double angles, such as \( \sin 2\theta \) and \( \cos 2\theta \). It is particularly useful when simplifying expressions involving double angles. The double angle identity for sine is:

• \( \sin 2\theta = 2 \sin \theta \cos \theta \)

In the original exercise, the left side was transformed to \( 1 - 2 \sin \theta \cos \theta \), which matches the form of the double angle identity when rearranged to the right side as \( 1 - \sin 2\theta \).
Many trigonometric problems can be simplified or solved using the double angle identity. It bridges single angle trigonometric expressions to double angle forms, making calculations or proofs more feasible.
Understanding and applying the double angle identity facilitates transformation of complex trigonometric expressions into simpler forms, ensuring the verification of various trigonometric identities.

Squaring Binomials

Squaring a binomial involves applying the formula:

• \((a - b)^2 = a^2 - 2ab + b^2\)

This is used when expressions like \( (\sin \theta - \cos \theta) \) are squared. Understanding this algebraic technique is crucial for simplifying trigonometric equations.
In the original exercise, squaring \( (\sin \theta - \cos \theta) \) resulted in \( \sin^2 \theta - 2 \sin \theta \cos \theta + \cos^2 \theta \). It shows the importance of correctly expanding the binomial first when verifying identities or solving equations.
Paying attention to each term during binomial expansion ensures accuracy in calculations, especially within the realm of trigonometry, where precise simplifications often lead to solutions.
Recognizing and properly expanding binomials set the foundation for further manipulation of trigonometric identities.