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Chapter 3: Problem 33

Factor each trigonometric expression. \(4 \sec ^{4} x-4 \sec ^{2} x+1\)

Short Answer

Expert verified
The expression factors to \( \left(2 \sec^{2} x - 1 \right)^2 \).

Step by step solution

01

Identify the quadratic form

Recognize that the expression can be written in a quadratic form in terms of \(\text{sec}^2 x\). Let \(y = \text{sec}^2 x\), and rewrite the expression: \(4y^2 - 4y + 1\).
02

Factor the quadratic expression

Factor \(4y^2 - 4y + 1\) using the quadratic formula or factoring techniques. This expression is a perfect square trinomial and can be factored as \((2y-1)^2\).
03

Substitute back \(y = \text{sec}^2 x\)

Replace \y\ with \text{sec}^2 x\ to get the factored form: \((2 \text{sec}^2 x - 1)^2\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Quadratic Form in Trigonometry
When tackling trigonometric expressions, sometimes they can be rewritten into a more familiar quadratic form. This process simplifies the factorization. For instance, in the expression \(4 \sec^{4} x - 4 \sec^{2} x + 1\), we can notice that \(\sec^{2} x\) appears in a way that resembles a quadratic equation.
To convert it into a quadratic form, we let \( y = \sec^{2} x \). This transforms the original expression to \(4y^2 - 4y + 1\). Now, it looks like a standard quadratic equation which is easier to handle.
Perfect Square Trinomial
A perfect square trinomial is a special form of a quadratic that can be factored into a binomial squared. Recognizing this form can greatly simplify factorization.
Consider the quadratic expression we obtained: \(4y^2 - 4y + 1\). This expression is indeed a perfect square trinomial.
A perfect square trinomial takes the form \( (ay + b)^2 \). In our case, \( 4y^2 - 4y + 1 = (2y - 1)^2 \). Once identified, this makes our factorization straightforward.
This factoring step is essential because it reveals the structure of the expression in a simpler form.
Trigonometric Substitution
After factoring the quadratic expression, we need to revert it back to its original trigonometric terms. This step is known as trigonometric substitution.
Initially, we set \( y = \sec^{2} x \), and after factoring, we obtained \( (2y - 1)^2 \). Substituting back, we replace \( y \) by \( \sec^{2} x \). Therefore, our expression becomes \( (2 \sec^{2} x - 1)^2 \).
This substitution reintroduces the trigonometric function \( \sec x \), and completes the factorization process. It鈥檚 a crucial final step to ensure the expression is in terms of the original variable \( x \).

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Most popular questions from this chapter

In each case, find \(\sin \alpha, \cos \alpha, \tan \alpha, \csc \alpha, \sec \alpha,\) and \(\cot \alpha\). \(\cos 2 \alpha=1 / 3\) and \(360^{\circ}<2 \alpha<450^{\circ}\)

For each equation, either prove that it is an identity or prove that it is not an identity. \(\tan \left(\frac{x}{2}\right)=\frac{1}{2} \tan x\)

Prove that each equation is an identity: $$ \cos ^{2}(A-B)-\cos ^{2}(A+B)=\sin ^{2}(A+B)-\sin ^{2}(A-B) $$

Suppose that \(\sin \alpha=1 / 4\) and \(\alpha\) is in quadrant II. Use identities to find the exact values of the other five trigonometric functions.

Simplify each expression by applying the odd/even identities, cofunction identities, and cosine of a sum or difference identities. Do not use a calculator $$ \sin \left(85^{\circ}\right) \sin \left(40^{\circ}\right)+\sin \left(-5^{\circ}\right) \sin \left(-50^{\circ}\right) $$
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