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Chapter 6: Problem 42

Verify each of the trigonometric identities. $$(\sec x+1)(\sec x-1)=\tan ^{2} x$$

Short Answer

Expert verified
The identity \((\sec x+1)(\sec x-1)=\tan ^{2} x\) is verified to be true.

Step by step solution

01

Expand the Left Side

We start by expanding the expression on the left side of the equation: \((\sec x + 1)(\sec x - 1)\). Using the difference of squares formula, \((a+b)(a-b) = a^2 - b^2\), where \(a = \sec x\) and \(b = 1\), we get: \[ \sec^2 x - 1. \]
02

Use the Pythagorean Identity

Recall the trigonometric identity: \[ \sec^2 x = 1 + \tan^2 x. \]Substitute this identity into our expression from Step 1:\[ \sec^2 x - 1 = (1 + \tan^2 x) - 1. \]
03

Simplify the Expression

Simplify the expression by subtracting 1:\[ (1 + \tan^2 x) - 1 = \tan^2 x. \]This confirms that both sides of the original equation are equal.

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Key Concepts

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

Understanding the Difference of Squares Formula
The difference of squares formula is a fundamental algebraic identity. It states that for any two terms, say \(a\) and \(b\), the expression \((a + b)(a - b)\) can be simplified to \(a^2 - b^2\).
This formula is extremely useful in factorization and simplification of polynomial expressions.
  • In trigonometry, this concept can be applied to simplify expressions involving trigonometric functions.
  • For example, in the exercise, \((\sec x + 1)(\sec x - 1)\) was transformed into \(\sec^2 x - 1\) by recognizing it as a difference of squares.
The simplicity and elegance of the formula make it an indispensable tool.
It allows us to break down complex expressions into more manageable parts.
Exploring the Pythagorean Identity
The Pythagorean identity is one of the cornerstone trigonometric identities.
It states that \(\sec^2 x = 1 + \tan^2 x\). This identity is derived from the basic Pythagorean theorem which states \(a^2 + b^2 = c^2\) for a right triangle.
In trigonometry:
  • The classic Pythagorean identity is \(\sin^2 x + \cos^2 x = 1\).
  • By dividing each term by \(\cos^2 x\), we derive \(\sec^2 x = 1 + \tan^2 x\).
This identity is useful because it allows us to relate secant and tangent functions.
In the solution, it was used to substitute \(\sec^2 x - 1\) with \(\tan^2 x\), thereby simplifying the expression.
Delving into the Secant Function
The secant function, denoted as \(\sec x\), is the reciprocal of the cosine function:
\[ \sec x = \frac{1}{\cos x}. \]
  • It is undefined when \(\cos x = 0\) since division by zero is impossible.
  • The secant function appears frequently in calculus, especially in the study of trigonometric integrals and derivatives.
Its role in trigonometry extends beyond being just a reciprocal.
It helps in understanding the relationships between different trigonometric functions.
In the given exercise, \(\sec x\) was part of the expression that demonstrated the difference of squares and the subsequent simplification using the Pythagorean identity.

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Most popular questions from this chapter

Determine whether each statement is true or false. $$\sin A \sin B=\sin A B$$

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

Monthly sales of soccer balls are approximated by \(S=400 \sin \left(\frac{\pi}{6} x\right)+2000,\) where \(x\) is the number of the month (January is \(x=1\), etc.). During which month do sales reach \(2400 ?\)

The number of deer on an island is given by \(D=200+100 \sin \left(\frac{\pi}{2} x\right),\) where \(x\) is the number of years since \(2000 .\) Which is the first year after 2000 that the number of deer reaches \(300 ?\)

Solve each trigonometric equation on \(0^{\circ} \leq \theta<360^{\circ} .\) Express solutions in degrees and round to two decimal places. $$\cos (2 x)+\frac{1}{2} \sin x=0$$
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