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Magazine Chapter 20: Problem 42
Prove the given identities.
$$1-\tan ^{2} x+\tan ^{4} x-\dots=\cos ^{2} x
\quad\left(-\frac{\pi}{4}
Short Answer
Expert verified
The series sums to \(\cos^2 x\).
Step by step solution
01
Recognize the Pattern
The given series is an infinite geometric series. It starts with 1, and the next terms are \(-\tan^2 x\), \(\tan^4 x\), and so forth. Recognize that these terms represent the series \(1 - \tan^2 x + (\tan^2 x)^2 - \dots\), where the first term \(a = 1\) and the common ratio \(r = -\tan^2 x\).
02
Sum of Infinite Series Formula
Recall the formula for the sum \(S\) of an infinite geometric series: \[ S = \frac{a}{1 - r} \]This formula applies when the absolute value of the common ratio \(|r|\) is less than 1.
03
Apply the Sum Formula
Apply the sum formula to our series. Here:- \(a = 1\) (the first term)- \(r = -\tan^2 x\) (the common ratio) Therefore, the sum \(S\) is:\[ S = \frac{1}{1 - (-\tan^2 x)} = \frac{1}{1 + \tan^2 x} \]
04
Use Trigonometric Identity
Utilize the Pythagorean identity for tangent and secant: \(1 + \tan^2 x = \sec^2 x\). Thus, \[ \frac{1}{1 + \tan^2 x} = \frac{1}{\sec^2 x} \] Recall that \(\sec x = \frac{1}{\cos x}\), so \(\sec^2 x = \frac{1}{\cos^2 x}\).
05
Simplify to Cosine
Simplify the expression:\[ \frac{1}{\sec^2 x} = \cos^2 x \]Thus proving that \( 1 - \tan^2 x + \tan^4 x - \dots = \cos^2 x \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Infinite Geometric Series
An infinite geometric series is a sum of terms where each term is derived from the previous term by multiplying by a constant known as the common ratio, denoted by \( r \). For such series, the sum becomes interesting when \/( |r| < 1 \/), as it allows the series to converge to a finite value. The series is represented as \( a + ar + ar^2 + ar^3 + \ldots \), where \( a \) is the first term. To find the sum of an infinite geometric series, we use the formula
- \( a = \text{initial term} \)
- \( S = \frac{a}{1 - r} \)
Sum Formula
The sum formula for an infinite geometric series is an incredibly useful tool for dealing with repetitive patterns in mathematics. When applied to a series like \( 1 - \tan^2 x + \tan^4 x - \dots \), understanding that each successive term differs only by the common ratio \( r = -\tan^2 x \) helps in analyzing the series.
- With \( a = 1 \) and \( r = -\tan^2 x \), we calculate the sum \( S \) using the formula: \( S = \frac{1}{1 + \tan^2 x} \).
Pythagorean Identity
The Pythagorean identity in trigonometry links the three fundamental trigonometric functions, providing a powerful method for simplifying expressions. The identity \( 1 + \tan^2 x = \sec^2 x \) is crucial in this context. It allows one to rewrite expressions in a more understandable form:
- \( \sec x = \frac{1}{\cos x} \)
- \( \sec^2 x = \frac{1}{\cos^2 x} \)
Tangent and Secant Functions
Understanding tangent and secant functions helps in bridging concepts when solving trigonometric problems. The tangent function, \( \tan x = \frac{\sin x}{\cos x} \), and the secant function, \( \sec x = \frac{1}{\cos x} \), are derivative of the core cosine function. They both play pivotal roles:
- Tangent measures the ratio of opposite to adjacent sides in a triangle.
- Secant translates cosine into multiplicative inverses, providing pathways from fractions to simpler expressions.
