Problem 41 - Precalculus: Functions and Graphs

Question

Verify the Identity. $$\left(1-\tan ^{2} \phi\right)^{2}=\sec ^{4} \phi-4 \tan ^{2} \phi$$

Step-by-Step Solution

Answer

The identity is verified: both sides simplify to \(1 - 2\tan^2 \phi + \tan^4 \phi\).

Step 1: Expand the Left Side

Start with the expression \((1 - \tan^2 \phi)^2\). Use the identity \((a - b)^2 = a^2 - 2ab + b^2\) to expand this as follows: \[ (1 - \tan^2 \phi)^2 = 1 - 2\tan^2 \phi + \tan^4 \phi \]

Step 2: Express the Right Side

The right side of the identity is \(\sec^4 \phi - 4 \tan^2 \phi\). We know from trigonometric identities that \(\sec^2 \phi = 1 + \tan^2 \phi\). So, \[ \sec^4 \phi = (\sec^2 \phi)^2 = (1 + \tan^2 \phi)^2 \]},

Step 3: Expand \(\sec^4 \phi\)

Now expand \((1 + \tan^2 \phi)^2\) using the formula for a square of a binomial \((a + b)^2 = a^2 + 2ab + b^2\): \[ (1 + \tan^2 \phi)^2 = 1 + 2\tan^2 \phi + \tan^4 \phi \] Substituting this back gives us the expression for the right side: \[ \sec^4 \phi - 4\tan^2 \phi = (1 + 2\tan^2 \phi + \tan^4 \phi) - 4\tan^2 \phi \]

Step 4: Simplify the Right Side

Simplify the expression: \[ 1 + 2\tan^2 \phi + \tan^4 \phi - 4\tan^2 \phi = 1 - 2\tan^2 \phi + \tan^4 \phi \]

Step 5: Compare Both Sides

Compare the simplified expression of the right side \(1 - 2 \tan^2 \phi + \tan^4 \phi\) with the expanded left side \(1 - 2\tan^2 \phi + \tan^4 \phi\). Both sides are equal, verifying the original identity.

Key Concepts

Expanding Binomials

When tackling problems involving binomials, expanding them can be a powerful tool. A binomial is a mathematical expression that has two terms, like \((a - b)^2\) or \((a + b)^2\). To expand a binomial squared, you will apply the formulas:

• For \((a + b)^2\): Ensure you remember it becomes \(a^2 + 2ab + b^2\).
• For \((a - b)^2\): This becomes \(a^2 - 2ab + b^2\).

By understanding these formulas, you simplify expressions and make them easier to work with in equations.
In this exercise, what we essentially did was taking \((1 - \tan^2 \phi)^2\) and expanding it using the formula for the square of a binomial. This step gives us a modified expression of \(1 - 2\tan^2 \phi + \tan^4 \phi\), which we can now use to further simplify and solve the identity.
Expanding binomials is about breaking down complex expressions into easier components, helping you verify or manipulate identities with ease.

Simplification Techniques

Simplification is another crucial aspect when verifying identities or solving mathematical problems. It's about reducing expressions down to their simplest form for easier comparison or further calculations.
When simplifying trigonometric identities, use known relationships between functions. In our example, the Pythagorean identity \(\sec^2 \phi = 1 + \tan^2 \phi\) was utilized to express \(\sec^4 \phi\) in terms of \(\tan^2 \phi\).
Here's a method to follow:

• Identify key identities or properties relevant to your problem. For trigonometric identities, these commonly include Pythagorean identities, complementary angle identities, and cofunction identities.
• Substitute to express more complex terms in simpler terms aligned with your identity goal.
• Manipulate and combine like terms to reach the lowest degree possible – here like \(\tan^2 \phi\) or constants.

By practicing and mastering these techniques, you will build a solid toolset for not only simplifying problems but verifying identities accurately.

Verifying Identities

Verifying trigonometric identities involves proving that two expressions are equivalent for all values of the involved variable. The process requires precision and an understanding of trigonometric properties.
In this context, we compared two expressions by transforming each into simplified forms and then checking their equality. For the presented problem, we started by expanding both sides and simplifying them until they matched.
Steps to verify identities effectively:

• First, decide which side of the equation to manipulate. Often, it's easier to work with the more complex side.
• Use algebraic manipulations like factoring or expanding to simplify each side.
• Apply appropriate identities to facilitate simplification. This might involve converting functions using formulas you already know.
• Once simplified, compare both sides. If they are identical, the identity is verified.

Verification is like solving a puzzle. It combines logic, algebraic methods, and trigonometric knowledge to prove that two sides of an equation are indeed the same.