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Chapter 5: Problem 43

Prove each of the following identities. $$ (\sin x-\cos x)^{2}=1-\sin 2 x $$

Short Answer

Expert verified
The identity \((\sin x - \cos x)^{2} = 1 - \sin 2x\) is proven by expanding, using identities, and simplifying both sides to the same expression.

Step by step solution

01

Expand the Left Side

Start by expanding \((\sin x - \cos x)^2\). This is a binomial expansion: \[(\sin x - \cos x)^2 = \sin^2 x - 2\sin x \cos x + \cos^2 x\]
02

Apply Pythagorean Identity

Use the Pythagorean identity \(\sin^2 x + \cos^2 x = 1\). Substitute into the expression to simplify it: \[\sin^2 x + \cos^2 x = 1 \quad \Rightarrow \quad \sin^2 x - 2\sin x \cos x + \cos^2 x = 1 - 2\sin x \cos x\]
03

Simplify the Right Side

Consider the right side of the identity, \(1 - \sin 2x\). Remember the double angle identity for sine: \[\sin 2x = 2\sin x \cos x\]Substitute to get:\[1 - \sin 2x = 1 - 2\sin x \cos x\]
04

Compare Both Sides

Both sides of the equation simplify to the same expression: Left Side: \(1 - 2\sin x \cos x\) Right Side: \(1 - 2\sin x \cos x\)Since both sides are equal, the identity is proven.

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Key Concepts

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

Pythagorean Identity
The Pythagorean Identity in trigonometry is a very powerful tool.It states that for any angle x: \\( \sin^2 x + \cos^2 x = 1 \).
This formula is derived from the Pythagorean Theorem.It represents a fundamental relationship between sine and cosine.
The identity comes from considering a right-angled triangle that is inscribed in the unit circle.The hypotenuse is the radius of the circle which has length 1.
  • \( \sin x \) is the y-coordinate on the unit circle.
  • \( \cos x \) is the x-coordinate on the unit circle.
  • Replacing these in the Pythagorean Theorem gives our identity: \( \sin^2 x + \cos^2 x = 1 \).

This identity is often used to simplify and prove trigonometric expressions.In the exercise, it helps replace \( \sin^2 x + \cos^2 x \) with 1 to simplify the expanded expression.It's the cornerstone for rearranging and simplifying many trigonometric problems.
Double Angle Identity
The Double Angle Identity for sine relates the sine of double an angle to sine and cosine of the single angle.For sine, the identity is given by: \\( \sin 2x = 2\sin x \cos x \).
This identity helps bridge trigonometric functions of larger angles to those of smaller angles.
The importance of this identity is apparent when proving identities or simplifying expressions involving multiple angles.It allows you to express the function in terms of simpler trigonometric functions which are easier to handle and manipulate.
  • Helps to understand oscillatory behaviors in functions.
  • Plays a vital role in problems involving angular motion.
  • Instrumental in making entire expressions easier to compare and simplify, as seen in our exercise when equating to \( 1 - 2\sin x \cos x \).

In our specific case, it equates the right side of the equation, ensuring we found the correct path to proving the identity.
Binomial Expansion
The Binomial Expansion is a formula to expand expressions raised to a power, such as \( (a - b)^2 \).Using this expansion gives us: \\((a - b)^2 = a^2 - 2ab + b^2 \).
This principle is not only crucial in algebra but also finds a place in trigonometry and calculus.
For trigonometric identities or functions, it often helps in breaking down complex expressions into more manageable parts.Thus, allowing us to apply other identities effectively, as it did in our exercise's first step.
  • Helps in breaking down complex algebraic and trigonometric operations.
  • Makes large computations simpler by expansion into smaller components.
  • Prepares ground for logical application of other identities, like Pythagorean or Double Angle identities.

In the initial step of our identity proof, using this expansion made it possible to apply the Pythagorean Identity, simplifying the complex expression to a recognizable form.

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Most popular questions from this chapter

Let \(\cos x=\frac{3}{4}\) with \(x\) in QIV and find the following. $$ \sin 2 x $$

Prove each of the following identities. $$ \frac{2-2 \cos 2 x}{\sin 2 x}=\sec x \csc x-\cot x+\tan x $$

The problems that follow review material we covered in Section \(4.3\). Graph one complete cycle. \(y=\cos \left(x-\frac{\pi}{3}\right)\)

Simplify each of the following. $$ 2 \cos ^{2} 105^{\circ}-1 $$

For Questions 4 through 6, determine if the statement is true or false. \(\cos 2 A=2 \cos A\)
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