Problem 21 - Precalculus: Functions and Graphs

Question

Verify the Identity. $$\sin ^{4} r-\cos ^{4} r=\sin ^{2} r-\cos ^{2} r$$

Step-by-Step Solution

Answer

The identity \(\sin^{4} r - \cos^{4} r = \sin^{2} r - \cos^{2} r\) is verified.

Step 1: Understand the Identity

The identity we need to verify is \(\sin^{4} r - \cos^{4} r = \sin^{2} r - \cos^{2} r\). We will use algebraic identities and trigonometric identities to verify it.

Step 2: Factor the Expression on the Left

Notice that \(a^4 - b^4\) can be factored using the difference of squares: \((a^2 - b^2)(a^2 + b^2)\). Thus, \(\sin^{4} r - \cos^{4} r = (\sin^{2} r - \cos^{2} r)(\sin^{2} r + \cos^{2} r)\).

Step 3: Use the Pythagorean Identity

From trigonometry, we know the Pythagorean identity: \(\sin^{2} r + \cos^{2} r = 1\). Substitute this into our expression: \((\sin^{2} r - \cos^{2} r)(\sin^{2} r + \cos^{2} r) = (\sin^{2} r - \cos^{2} r)(1)\).

Step 4: Simplify the Expression

Since multiplying by 1 does not change an expression, we have \((\sin^{2} r - \cos^{2} r)(1) = \sin^{2} r - \cos^{2} r\). Thus, the left side simplifies exactly to the right side.

Step 5: Conclusion

Both sides of the original equation \(\sin^{4} r - \cos^{4} r = \sin^{2} r - \cos^{2} r\) are equal, confirming that the identity is verified.

Key Concepts

Pythagorean Identity

The Pythagorean identity is a fundamental trigonometric identity that is immensely helpful in simplifying expressions and verifying other identities. It states:

• \(\sin^2 r + \cos^2 r = 1\)

This identity is derived from the Pythagorean theorem in geometry, which relates the sides of a right triangle to its hypotenuse. Think of a unit circle on a coordinate plane, where a line from the center to the perimeter forms the hypotenuse of a right triangle.
The horizontal and vertical components can be described using sine and cosine, which when squared and summed, always give one. This property is crucial because it allows us to transform and simplify trigonometric expressions by substituting \(\sin^2 r + \cos^2 r\) with 1.
In our original exercise, employing the Pythagorean identity allows us to simplify \((\sin^2 r + \cos^2 r)\) to 1. This simplification is key in proving the given identity \(\sin^{4} r - \cos^{4} r = \sin^{2} r - \cos^{2} r\).

Difference of Squares

The difference of squares is a powerful algebraic identity that helps factorise expressions. It states that for any two numbers \(a\) and \(b\):

• \(a^2 - b^2 = (a - b)(a + b)\)

This identity is used to break down complex algebraic expressions into simpler, more manageable factors. It works on the principle that the square of a number multiplied by itself can break into two binomial factors.
In our exercise, this concept is extended to factor the expression \(\sin^{4} r - \cos^{4} r\). Recognizing \(a^4 - b^4\) as \((a^2 - b^2)(a^2 + b^2)\), we can set \(a = \sin^2 r\) and \(b = \cos^2 r\). This results in:

• \(\sin^{4} r - \cos^{4} r = (\sin^2 r - \cos^2 r)(\sin^2 r + \cos^2 r)\)

Using this approach simplifies the process of verifying our trigonometric identity.

Algebraic Identities

Algebraic identities are fundamental properties that help in simplifying and manipulating algebraic expressions. They are known truths that don’t change, no matter what values you substitute into the variables. The basic algebraic identities include:

• \((a + b)^2 = a^2 + 2ab + b^2\)
• \((a - b)^2 = a^2 - 2ab + b^2\)
• \(a^2 - b^2 = (a - b)(a + b)\)

These identities are essential tools for factorization and simplifying equations. Recognizing patterns and substituting them using these identities can make solving equations far easier.
In our original exercise, using the difference of squares is a direct application of algebraic identities. Recognizing \(\sin^{4} r - \cos^{4} r\) as \(a^4 - b^4\), we were able to employ the identity to write it in factored form.
This step was pivotal in confirming \(\sin^{4} r - \cos^{4} r = \sin^{2} r - \cos^{2} r\) by showing the expression was indeed equal on both sides of the equation.