Problem 25 - Precalculus with Calculus Previews

Question

Verify the given identity. $$ 1-\cos ^{4} \theta=\left(2-\sin ^{2} \theta\right) \sin ^{2} \theta $$

Step-by-Step Solution

Answer

The identity is verified: both sides simplify to \(2\sin^2 \theta - \sin^4 \theta\).

Step 1: Simplify the Left-Hand Side

Start with the left-hand side (LHS) of the given identity: \(1 - \cos^4 \theta\). Notice that this can be rewritten using the identity \(a^2 - b^2 = (a-b)(a+b)\) as:\[1 - \cos^4 \theta = (1 + \cos^2 \theta)(1 - \cos^2 \theta)\]The term \(1 - \cos^2 \theta\) is equal to \(\sin^2 \theta\) based on the Pythagorean identity \( \sin^2 \theta + \cos^2 \theta = 1 \). Thus:\[1 - \cos^4 \theta = (1 + \cos^2 \theta) \sin^2 \theta\]

Step 2: Simplify the Expression

Now, substitute \(\sin^2 \theta\) for \(1 - \cos^2 \theta\) in the LHS:\[ (1 + \cos^2 \theta) \sin^2 \theta \]Expand this expression to verify if it matches the right-hand side (RHS):\[ \sin^2 \theta + \cos^2 \theta \sin^2 \theta \]

Step 3: Verify the Right-Hand Side

The RHS of the given identity is \((2 - \sin^2 \theta) \sin^2 \theta\).Multiply to expand the RHS:\[ 2\sin^2 \theta - \sin^4 \theta \]

Step 4: Compare Both Sides

Compare the expanded expressions:LHS: \[ \sin^2 \theta + \cos^2 \theta \sin^2 \theta \]RHS: \[ 2\sin^2 \theta - \sin^4 \theta \]Rewrite \(\cos^2 \theta \sin^2 \theta\) using \(\cos^2 \theta = 1 - \sin^2 \theta\):\[ \sin^2 \theta + (1 - \sin^2 \theta) \sin^2 \theta = \sin^2 \theta + \sin^2 \theta - \sin^4 \theta \]Which simplifies to \[ 2\sin^2 \theta - \sin^4 \theta \], the same as the RHS. Both sides match, verifying the given identity.

Key Concepts

Pythagorean Identity

The Pythagorean Identity is one of the fundamental trigonometric identities. It states that for any angle \( \theta \), the equation \( \sin^2 \theta + \cos^2 \theta = 1 \) holds true. This identity is derived from the Pythagorean theorem, and it's foundational in trigonometry because it relates the sine and cosine of an angle to unity. It allows us to express sine in terms of cosine and vice versa.

For example, if we know \( \cos^2 \theta \), we can find \( \sin^2 \theta \) using the formula \( \sin^2 \theta = 1 - \cos^2 \theta \). In our exercise, this identity was used to simplify \( 1 - \cos^2 \theta \) to \( \sin^2 \theta \), providing a critical step in verifying the given trigonometric identity.

Understanding and applying the Pythagorean Identity is crucial when solving and simplifying various trigonometric problems. It often acts as the bridge between different forms of expressions.

Simplifying Expressions

Simplifying expressions is a process where we transform a complex equation into a simpler or more recognizable form. This makes it easier to manage and understand. In the context of trigonometry, it often involves using identities such as the Pythagorean Identity to reduce expressions.

In our solution, we started with the left-hand side, which was \( 1 - \cos^4 \theta \). By recognizing that this is a difference of squares, we rewrote it as \((1 + \cos^2 \theta)(1 - \cos^2 \theta)\). Then, using the Pythagorean Identity, we further simplified \(1 - \cos^2 \theta\) to \(\sin^2 \theta\). Thus, the expression became \((1 + \cos^2 \theta) \sin^2 \theta\).

By breaking complex expressions into products or sums of trigonometric functions and identities, we make it easier to compare sides of an equation, which is a useful strategy in verifying identities. Simplifying is a valuable skill, especially in trigonometry, where the expressions can often look complicated at first glance.

Verifying Identities

Verifying identities involves showing that two different trigonometric expressions are equivalent, meaning they have the same value for all valid inputs of the variables involved. This process usually requires manipulation and simplification of one or both sides of the given equation using known identities and algebraic techniques.

In the given problem, we were tasked to verify the identity \(1-\cos ^{4} \theta = (2 - \sin^2 \theta) \sin^2 \theta\). We independently simplified both the left-hand side (LHS) and the right-hand side (RHS) using algebra and trigonometric identities. The goal was to rewrite both sides of the equation in the same form, proving their equivalence.

* On the LHS, after simplification, we arrived at \(\sin^2 \theta + \cos^2 \theta \sin^2 \theta\) and then further refined it to \(2\sin^2 \theta - \sin^4 \theta\).* The RHS, when expanded, directly simplified to \(2\sin^2 \theta - \sin^4 \theta\).

Since both sides simplified to the same expression, the identity was verified. Verifying such identities helps deepen understanding of trigonometric concepts by confirming that different approaches lead to the same trigonometric result.