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Chapter 6: Problem 39

Verify each of the trigonometric identities. $$(\sin x+\cos x)^{2}+(\sin x-\cos x)^{2}=$$

Short Answer

Expert verified
The identity simplifies to 2.

Step by step solution

01

Expand the squares

Start by expanding each square in the expression separately: \[(\sin x + \cos x)^2 = \sin^2 x + 2\sin x \cos x + \cos^2 x\] \[(\sin x - \cos x)^2 = \sin^2 x - 2\sin x \cos x + \cos^2 x\]
02

Combine the expanded expressions

Add the two expanded expressions together: \[\sin^2 x + 2\sin x \cos x + \cos^2 x + \sin^2 x - 2\sin x \cos x + \cos^2 x\]
03

Simplify the expression

Combine like terms from the expression:- \(2\sin x \cos x\) and \(-2\sin x \cos x\) cancel each other out.- Combine \(\sin^2 x + \cos^2 x + \sin^2 x + \cos^2 x\) to result in \(2(\sin^2 x + \cos^2 x)\).
04

Apply the Pythagorean identity

Use the Pythagorean identity \(\sin^2 x + \cos^2 x = 1\) to simplify further:\[2(\sin^2 x + \cos^2 x) = 2 imes 1 = 2\]

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Key Concepts

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

Expanding Squares
When faced with an expression like \((\sin x + \cos x)^{2} + (\sin x - \cos x)^{2}\), the first step involves expanding each square. This process is similar to the algebraic identity \((a + b)^2 = a^2 + 2ab + b^2\). The same rules apply when dealing with trigonometric terms:
  • \((\sin x + \cos x)^2\) expands to \(\sin^2 x + 2\sin x \cos x + \cos^2 x\).
  • Similarly, \((\sin x - \cos x)^2\) expands to \(\sin^2 x - 2\sin x \cos x + \cos^2 x\).
By thoroughly expanding the squares, we lay the groundwork for further simplification. Each term must be accurately computed to ensure the integrity of the identities being verified.
Pythagorean Identity
The Pythagorean identity \(\sin^2 x + \cos^2 x = 1\) is a fundamental concept in trigonometry. It proves crucial when simplifying expressions involving sine and cosine terms. After expanding the squares and combining like terms, we usually end up with terms like \(\sin^2 x + \cos^2 x\).This is where the Pythagorean identity comes into play. Instead of handling the separate terms of sine and cosine squares each time, we can replace them with 1, making expressions tidier and more manageable. In practice, this identity simplifies calculations and is essential when verifying trigonometric equations.
Simplification of Expressions
Once individual terms are expanded and identities such as the Pythagorean identity are applied, it becomes time to simplify the expression. The concept of simplification is all about making the expression as straightforward as possible. To achieve this:
  • First, combine like terms. For example, in our exercise, the terms \(+2\sin x \cos x\) and \(-2\sin x \cos x\) cancel each other out.
  • Next, use the Pythagorean identity to replace \(\sin^2 x + \cos^2 x\) with 1, then multiply by any coefficient outside the parentheses.
After these steps, we often arrive at a much simpler, minimal expression. Simplification not only checks correctness but also helps in gaining deeper insights into the behavior of the function.

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Most popular questions from this chapter

Find the smallest positive values of \(x\) that make the statement true. Give the answer in degrees and round to two decimal places. $$2 \sin x=\tan x,-\frac{\pi}{3} \leq x \leq \frac{\pi}{3}$$

Write \(\cos A \cos B \cos C\) as a sum or difference of sines and cosines.

Solve for the smallest positive \(x\) that makes this statement true: $$\cos x \cos 15^{\circ}+\sin x \sin 15^{\circ}=0.7$$

Find the smallest positive values of \(x\) that make the statement true. Give the answer in degrees and round to two decimal places. $$\cos x=2-\cos x, 0 \leq x \leq 2 \pi$$

Find the smallest positive values of \(x\) that make the statement true. Give the answer in degrees and round to two decimal places. $$e^{x}-\tan x=0$$
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