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Chapter 20: Problem 33

Prove the given identities. $$\cos ^{2} x-\sin ^{2} x=1-2 \sin ^{2} x$$

Short Answer

Expert verified
The identity is proven using the double-angle identity for cosine: \( \cos(2x) = \cos^2 x - \sin^2 x = 1 - 2\sin^2 x \).

Step by step solution

01

Start from the left-hand side (LHS) of the Identity

The left-hand side of the given identity is \( \cos^2 x - \sin^2 x \). Recognize that this is a form of the double-angle identity for cosine: \( \cos(2x) = \cos^2 x - \sin^2 x \). Thus, we can write the LHS as \( \cos(2x) \).
02

Transform Using Double-Angle Identity for Cosine

Use the double-angle identity \( \cos(2x) = 1 - 2\sin^2 x \) directly on the LHS. Thus, \( \cos^2 x - \sin^2 x = 1 - 2\sin^2 x \) according to the double-angle formula for cosine.
03

Compare with the Right-Hand Side (RHS)

The right-hand side is \( 1 - 2\sin^2 x \). Note that this matches the expression we derived in Step 2 from the LHS using the double-angle identity. Therefore, both sides of the equation are equal.

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Key Concepts

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

Double-Angle Formulas
The Double-Angle Formulas are essential in trigonometry. They allow us to express trigonometric functions of double angles in terms of single angles. Among these, the formula for cosine is particularly useful: \[\cos(2x) = \cos^2(x) - \sin^2(x)\] This shows how the cosine of twice an angle relates directly to both the cosine and sine of the original angle. Understanding this identity helps simplify complex trigonometric expressions and solve equations involving periodic functions.
  • Key Insight: The formula combines both cosine squared and sine squared terms, making it versatile for trigonometric manipulations.
  • Uses: It's used extensively in proofs, such as showing equivalencies between different trigonometric identities, and is a foundational tool in calculus problems involving integration and derivatives of trigonometric functions.
Knowing how to use these formulas effectively is a critical skill in engineering, physics, and mathematics.
Cosine Identity
The Cosine Identity, particularly the double-angle form we discussed, plays a central role in transforming trigonometric expressions. The identity provided in the problem, \[\cos^2(x) - \sin^2(x) = \cos(2x)\] helps us convert the squared terms into a single trigonometric function. This is beneficial when working with expressions that involve multiple angles or when integrating and differentiating trigonometric functions.
  • Simplification: By recognising the form of the identity, complex expressions can be greatly simplified.
  • Computation: It simplifies the calculations, especially when proving identities or evaluating trigonometric functions for specific angles.
In essence, the cosine identity provides a bridge to link different trigonometric concepts together, offering a new perspective on problem-solving.
Sine Squared Identity
The Sine Squared Identity is another cornerstone in understanding trigonometric relationships. It refers to how sine squares are interconnected with other trigonometric functions. The identity \[1 - \sin^2(x) = \cos^2(x)\] reveals the inherent relationship between sine and cosine. When we look at the double-angle formula for cosine: \[\cos(2x) = 1 - 2\sin^2(x)\] this reveals how sine squared transforms and contributes to forming the double-angle identities.
  • Applications: Frequently used in calculus, this identity helps in solving differential equations and computing integrals involving sine and cosine terms.
  • Visual Insight: It provides insights into how the unit circle functions. Since \(\cos^2(x) + \sin^2(x) = 1\), this identity continuously ties back to the Pythagorean identity, adding depth to our understanding of trigonometry.
Mastering these relationships is key to effectively tackling higher-level math problems.

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Most popular questions from this chapter

Solve the given equations graphically. $$\sqrt{x}-\sin 3 x=1$$

Evaluate each expression by first changing the form. Verify each by use of a calculator. $$\sin 122^{\circ} \cos 32^{\circ}-\cos 122^{\circ} \sin 32^{\circ}$$

Use the half-angle formulas to solve the given problems. In electronics, in order to find the root-mean-square current in a circuit, it is necessary to express \(\sin ^{2} \omega t\) in terms of \(\cos 2 \omega t .\) Show how this is done.

Prove the given identities. $$\sin (x+y) \sin (x-y)=\sin ^{2} x-\sin ^{2} y$$

Use a calculator to determine whether the given equations are identities. $$\frac{2 \cos ^{2} x-1}{\sin x \cos x}=\tan x-\cot x$$
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