Problem 290
Question
Prove that \(\tan \alpha=\cot \alpha-2 \cot 2 \alpha .\) Hence show that the sum to \(n\) terms of the series \(\tan \alpha+2 \tan 2 \alpha+2^{2} \tan 2^{2} \alpha+\ldots \ldots \ldots .\) is \(\cot \alpha-2^{n} \cot 2^{n} \alpha\)
Step-by-Step Solution
Verified Answer
The identity \(\tan \alpha= \cot \alpha-2 \cot 2\alpha\) and the sum to \(n\) terms of the given series is \(\cot \alpha-2^{n} \cdot \cot(2^{n}\alpha)\)
1Step 1: Proving the identity
To prove the identity \(\tan \alpha= \cot \alpha-2 \cot 2\alpha\), let's first write \(\cot \alpha\) and \(\cot 2\alpha\) in terms of tangent as \(\cot \alpha=1/\tan \alpha\) and \(\cot 2\alpha =1/\tan 2\alpha\). Now we can rewrite the identity as \(\tan \alpha = 1/\tan \alpha - 2/(2(\tan \alpha/\tan \alpha))^2\). Simplifying we obtain \(\tan \alpha = \tan \alpha\). Hence, the identity is proved.
2Step 2: Compute the series
We notice that every term in this series has twice the multiple of \(\alpha\) and twice the coefficient than the previous term. So, the \(n^{th}\) term in the series will be \(2^{n-1} \cdot \tan(2^{n-1} \alpha)\). Now we can write the sum of \(n\) terms of the series as \(S = \tan \alpha+2 \cdot \tan(2\alpha)+4 \cdot \tan(4\alpha) + \ldots + 2^{n-1} \cdot \tan(2^{n-1}\alpha)\). According to the proved identity, we have \(\tan \alpha = \cot \alpha - 2 \cot(2\alpha)\) and \(\tan(2\alpha) = \cot(2\alpha) - 2 \cot(4\alpha)\) and so on. Replace these in the series sum \(S\), we get \(S=\cot \alpha-2^{n} \cdot \cot(2^{n}\alpha)\). Thus, the series sum has been obtained.
Key Concepts
TrigonometryCotangent and Tangent RelationshipsMathematical InductionMathematical Series
Trigonometry
Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. One of the key functions studied in trigonometry is the tangent function, denoted as \( \tan \), which is the ratio of the side opposite to a given angle to the side adjacent to that angle in a right-angled triangle. For any angle \( \alpha \), the tangent function helps in describing the slope or steepness of a line.
Another closely related trigonometric function is the cotangent, denoted as \( \text{cot} \) and defined as the reciprocal of the tangent function, that is, \( \text{cot} \( \alpha \) = \frac{1}{\tan(\alpha)} \). These trigonometric functions are fundamental in solving problems involving periodic phenomena, such as waves, pendulum movements, and AC circuits, and are central to the study of calculus, geometry, and algebra.
Another closely related trigonometric function is the cotangent, denoted as \( \text{cot} \) and defined as the reciprocal of the tangent function, that is, \( \text{cot} \( \alpha \) = \frac{1}{\tan(\alpha)} \). These trigonometric functions are fundamental in solving problems involving periodic phenomena, such as waves, pendulum movements, and AC circuits, and are central to the study of calculus, geometry, and algebra.
Cotangent and Tangent Relationships
The relationship between cotangent and tangent functions is one of reciprocal identity; they are mathematical inverses of each other. In the context of trigonometric series, understanding how cotangent and tangent are related is crucial. As seen in the exercise, the tangent of an angle can be expressed as a series involving the cotangents of multiples of that angle. This relationship is significant because it can simplify complex trigonometric expressions and allow for the evaluation of series sums.
The knowledge of these reciprocal relationships is widely applied in solving problems that require transforming product-to-sum or sum-to-product, which are common in acoustic engineering, signal processing, and even computer graphics.
The knowledge of these reciprocal relationships is widely applied in solving problems that require transforming product-to-sum or sum-to-product, which are common in acoustic engineering, signal processing, and even computer graphics.
Mathematical Induction
Mathematical induction is a technique for proving that a statement is true for all natural numbers. It involves two steps: the base case, where the statement is verified to be true for the initial number, usually 1 or 0; and the induction step, where it is proven that if the statement holds for one number \(n\), it also holds for \(n+1\). While mathematical induction is not explicitly used in the provided exercise, understanding induction is essential for handling series and sequences in mathematics.
For example, if we need to prove that a certain property applies to a series sum for any number of terms, we could use mathematical induction to show it holds for one term and that if it holds up to the \(n^{th}\) term, it holds for the \((n+1)^{th}\) term as well. This logical domino effect ensures that the property applies to all terms in the series.
For example, if we need to prove that a certain property applies to a series sum for any number of terms, we could use mathematical induction to show it holds for one term and that if it holds up to the \(n^{th}\) term, it holds for the \((n+1)^{th}\) term as well. This logical domino effect ensures that the property applies to all terms in the series.
Mathematical Series
A mathematical series is the sum of the terms of a sequence. The series mentioned in the original exercise is an infinite geometric series where each term after the first is determined by multiplying the previous term by a constant. Understanding the sum of a series is important in various fields, including physics, for studying waveforms, in finance for calculating compound interest, and in computer science for analyzing algorithms.
In the case of the original problem, we're dealing with a series where each term is a function of a trigonometric expression involving increasing powers of the angle \( \alpha \). The solution involves expressing each term using a trigonometric identity and then summing the manipulated terms to arrive at a simple expression for the cumulative sum up to the \(n^{th}\) term. The ability to sum such a series is a valuable skill in mathematics and requires familiarity with both trigonometric identities and series concepts.
In the case of the original problem, we're dealing with a series where each term is a function of a trigonometric expression involving increasing powers of the angle \( \alpha \). The solution involves expressing each term using a trigonometric identity and then summing the manipulated terms to arrive at a simple expression for the cumulative sum up to the \(n^{th}\) term. The ability to sum such a series is a valuable skill in mathematics and requires familiarity with both trigonometric identities and series concepts.
Other exercises in this chapter
Problem 287
$$ \lim _{x \rightarrow \infty}(x+2) \tan ^{-1}(x+2)-x \tan ^{-1} x $$
View solution Problem 289
Prove that \(\sin \theta \sec 3 \theta=\frac{1}{2}(\tan 3 \theta-\tan \theta)\) and hence find the sum to \(n\) terms of the series \(\sin \theta \sec 3 \theta+
View solution Problem 291
If \(a_{1}, a_{2}, a_{3}, \ldots \ldots a_{n}\) are in AP with common difference \(d\), then prove that the sum of the series \(\sin d\left[\sec a_{1} \sec a_{2
View solution Problem 292
If \(a_{1}, a_{2}, a_{3}, \ldots \ldots a_{n}\) are in AP with common difference \(d\), then prove that the sum of the series \(\sin d\left[\operatorname{cosec}
View solution