Problem 104 - Algebra and Trigonometry Real Mathematics, Real People

Question

Verify the identity algebraically. Use a graphing utility to check your result graphically. $$\cos ^{4} x-\sin ^{4} x=\cos 2 x$$

Step-by-Step Solution

Answer

The requested trigonometric identity \(\cos ^{4} x-\sin ^{4} x=\cos 2 x\) is correctly verified both algebraically and graphically.

Step 1: Rewrite Using Power-Reducing Formula

We are given the identity \( \cos ^{4} x-\sin ^{4} x \). This equation looks a lot like the power-reducing formula \((a^4 - b^4) = (a^2 - b^2)^2\), so rewrite it as \( (\cos^2x - \sin^2x)^2 \).

Step 2: Use Double-Angle Formula

The double-angle formula states that \( \cos(2x) = \cos^2(x) - \sin^2(x) \). Therefore, we replace \(\cos^2x - \sin^2x\) with \( \cos(2x) \) in the equation, so the equation becomes \( (\cos(2x))^2 \).

Step 3: Simplify

Squaring any real number, including \( \cos(2x) \), always gives a positive number or zero. So, \( (\cos(2x))^2 = \cos^2(2x) \).

Step 4: Graphing

To confirm the results graphically, plot \(\cos ^{4} x-\sin ^{4} x\) and \( \cos(2x) \) in the same graphing utility and confirm that they coincide.

Key Concepts

Power-Reducing Formulas

Power-reducing formulas are crucial in trigonometry for simplifying expressions containing squared trigonometric functions. These formulas allow us to express powers of trigonometric functions in terms of the first power. This is especially useful for integration, simplification, or converting expressions to simpler forms.

• The formula for reducing the power of sine is: \( \sin^2(x) = \frac{1 - \cos(2x)}{2} \)
• For cosine, it is: \( \cos^2(x) = \frac{1 + \cos(2x)}{2} \)

These formulas not only simplify calculations but also relate the trigonometric function to its double angle. This connection is the key when verifying identities or solving equations, just like turning \( \cos^4x - \sin^4x \) into something more workable. By rewriting such high power expressions, the complexity of problems is reduced to more manageable terms.

Double-Angle Formulas

Double-angle formulas are vital tools for working with trigonometric equations. They allow you to express trigonometric functions of doubled angles in terms of single angles, facilitating the simplification and solution of trigonometric expressions.

• The formula for cosine is: \( \cos(2x) = \cos^2(x) - \sin^2(x) \)
• Another version of this formula is: \( \cos(2x) = 2\cos^2(x) - 1 \)
• And yet another is: \( \cos(2x) = 1 - 2\sin^2(x) \)

In our exercise, we used the identity \( \cos(2x) = \cos^2(x) - \sin^2(x) \) to directly replace \( \cos^2x - \sin^2x \) in the original expression \( \cos^4 x - \sin^4 x \). The translation of a complex equation into one involving a double angle dramatically eases the verification of trigonometric identities.

Graphing Utilities

Graphing utilities, such as graphing calculators or software, are incredibly useful in confirming algebraic solutions by visual checking. They allow us to see trigonometric identities graphically, providing a visual representation to verify solutions.To use a graphing utility effectively:

• Enter the expressions you want to compare as separate equations. For our exercise, this was \( \cos^4 x - \sin^4 x \) and \( \cos(2x) \).
• Graph each equation on the same set of axes to determine if they coincide or overlap perfectly.
• If the graphs coincide over the interval of interest, your algebraic solution is confirmed visually.

Using graphing utilities gives a different perspective to solve trigonometry problems, combining analytical proofs with graphical insights for robust verification. This method is especially helpful for visual learners or when you need additional confirmation beyond algebraic proof.