Problem 156 - Precalculus

Question

For the following exercises, prove the identities. $$ \left(\sin ^{2} x-1\right)^{2}=\cos (2 x)+\sin ^{4} x $$

Step-by-Step Solution

Answer

The identity is proven: both sides simplify to \( \cos^4 x \).

Step 1: Understanding the Identity

We are given the identity \( (\sin^2 x - 1)^2 = \cos(2x) + \sin^4 x \). Our goal is to manipulate one or both sides of the equation using trigonometric identities to prove both expressions are equivalent.

Step 2: Simplifying the Left Side

Start with the left side of the equation: \[ (\sin^2 x - 1)^2 \]Use the identity \( 1 - \sin^2 x = \cos^2 x \) to rewrite:\[ (\sin^2 x - 1)^2 = (-(1 - \sin^2 x))^2 = (-\cos^2 x)^2 = \cos^4 x \].

Step 3: Simplifying the Right Side

Now, let's examine the right side:\[ \cos(2x) + \sin^4 x \]Using the identity \( \cos(2x) = 2\cos^2 x - 1 \), rewrite the expression:\[ 2\cos^2 x - 1 + \sin^4 x \].

Step 4: Proving the Identity

Now let's compare both sides:The expression on the left side simplifies to \( \cos^4 x \).On the right, use the identity \( \sin^2 x = 1 - \cos^2 x \) to express \( \sin^4 x \) as:\[ \sin^4 x = (\sin^2 x)^2 = (1 - \cos^2 x)^2 \]Plug this back into the right side expression:\[ 2\cos^2 x - 1 + (1 - \cos^2 x)^2 \]Simplify by expanding \[ = 2\cos^2 x - 1 + (1 - 2\cos^2 x + \cos^4 x) = \cos^4 x + \cancel{2\cos^2 x} - 2\cos^2 x + \cancel{2\cos^2 x} - 1 + 1 = \cos^4 x\].Both sides are equal, thus proving the identity.

Key Concepts

Pythagorean Identity

The Pythagorean identity is one of the most fundamental identities in trigonometry, connecting sine and cosine functions. It is expressed as:

• \( \sin^2 x + \cos^2 x = 1 \)

This equation is derived from the Pythagorean theorem and applies to any angle \( x \). It helps us express one trigonometric function in terms of another.
For example, we can rearrange this identity to solve for \( \sin^2 x \) as \( \sin^2 x = 1 - \cos^2 x \). Alternatively, we could solve for \( \cos^2 x \) as \( \cos^2 x = 1 - \sin^2 x \).
These rearrangements are crucial in simplifying expressions and proving trigonometric identities, just as we saw in the given exercise.

Double Angle Formula

The double angle formula is another essential tool in trigonometry. It allows us to express trigonometric functions of double angles, such as \( \cos(2x) \), in terms of functions at a single angle \( x \).
For cosine, the double angle formula is:

• \( \cos(2x) = 2\cos^2 x - 1 \)

This formula is highly useful for reducing complex expressions or transforming one trigonometric function into another.
In our exercise, the double angle formula of cosine helped to rewrite and simplify the expression on the right side of the equation by breaking it into pieces that can be further simplified or squared.

Squared Trigonometric Functions

Squared trigonometric functions like \( \sin^2 x \) and \( \cos^2 x \) are frequently used for identity proofs and simplifications. When trigonometric functions are squared, they often form part of expressions linked with the Pythagorean identity.
For example, in the exercise, \( \sin^4 x \) was rearranged by recognizing it as \((\sin^2 x)^2\). Using \( \sin^2 x = 1 - \cos^2 x\), this becomes \((1 - \cos^2 x)^2\).
By squaring \(1 - \cos^2 x\), the expression can go through expansion and simplification, similar to how polynomials are treated. This assists in equating both sides of the given identity.

Trigonometric Equation Solving

Solving trigonometric equations involves using identities and formulas to manipulate expressions until both sides of an equation match or reduce to a known identity. The process often requires a series of simplifications and substitutions, leveraging identities like the Pythagorean identity and formulas like the double angle formula.
In the given exercise, the equation was solved by proving an identity. The solution required:

• Simplifying either side of the equation using appropriate identities.
• Recognizing terms that can be rewritten or simplified further.
• Carefully expanding squared terms and combining like terms.

The goal was to show that both sides of the equation represented the same expression, ultimately demonstrating the trigonometric identity was true. This approach is essential in establishing truths within trigonometry and enhances understanding of how trigonometric functions interact in equations.