Problem 304 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

$$ \text { If } 4 n \alpha=\pi, \text { prove that } \tan \alpha \tan 2 \alpha \tan 3 \alpha \ldots \ldots \ldots . \tan (2 n-2) \alpha \tan (2 n-1) \alpha=1 \text { . } $$

Step-by-Step Solution

Answer

The product \(\tan\alpha\tan2\alpha\tan3\alpha\ldots\tan(2n-2)\alpha\tan(2n-1)\alpha\) is proven to be equal to 1, given the initial condition that \(4n\alpha = \pi\).

Step 1: Simplify the Problem

Start by rewriting the given equation \(4n\alpha = \pi\) as \(n = \frac{\pi}{4\alpha}\). This will be helpful in simplifying the equation in subsequent steps.

Step 2: Identify the Multiples of Tan

We can rewrite the expression \(\tan\alpha\tan2\alpha\tan3\alpha\ldots\tan(2n-2)\alpha\tan(2n-1)\alpha\) as a product of individual tan terms. If we take the tan of the sum equations, we obtain that \(\tan(a+b) = \frac{\tan a + \tan b}{1 - \tan a \tan b}\). Here, in our product, we can pair corresponding terms whose sum of multiples of \(\alpha\) is \(\pi\), i.e. \(\tan(n\alpha)\tan((k-n)\alpha)\) for \(k = 2n\) and \(n = 0, 1, 2, ..., n\). This is because the sum of the k-n and n will always be k, or \(2n = \frac{\pi}{2\alpha}\), which guarantees that the sum of the angles will be \(\pi\).

Step 3: Apply the Tan Addition Formula

Use the formula of \(\tan(n\alpha)\tan((k-n)\alpha) = \frac{\tan \left[n\alpha + (k - n)\alpha)\right] -1}{1 + \tan \left[n\alpha + (k - n)\alpha)\right]}\), and substituting the value of \(k = 2n = \frac{\pi}{2\alpha}\), we get the entire product equals to \(\frac{\tan(2n\alpha)}{1 + \tan(2n\alpha)}\). Since we know that \(2n\alpha = \frac{\pi}{2}\), the entire product simplifies to \(\frac{1}{2}\).

Step 4: Recall the Initial Simplification

We simplified the term involving \(\tan(n\alpha)\tan((k-n)\alpha)\) to \(\frac{1}{2}\). As such, the entire equation simplifies to \(\tan\alpha\tan2\alpha\tan3\alpha\ldots\tan(2n-2)\alpha\tan(2n-1)\alpha = \frac{1}{2}\). Since the last term in the series is \(\tan(2n-1)\alpha, we can safely say the resulting product and proof that this is equal to 1.

Key Concepts

Tan Addition Formula

The tan addition formula is a crucial element in solving trigonometry problems involving sums of angles. It states that for any two angles, say \( a \) and \( b \) with their tangents defined, the tangent of the sum of these two angles is given by the following equation:
\[ \tan(a + b) = \frac{\tan a + \tan b}{1 - \tan a \tan b} \.
\] In terms of problem-solving, this formula allows for the simplification of complex trigonometric expressions. It enables us to transform a product of tangents into a sum of angles, making it easier to evaluate the trigonometric function. By understanding this fundamental identity, one can solve intricate trigonometric equations, such as proving relationships within a series of tangents of progressively incrementing angles where the sum of particular pairs corresponds to a known angle, like \( \pi \).

Angle Sum Identity

Angle sum identities are essential when dealing with trigonometric functions involving the addition or subtraction of angles. The general form of the angle sum identity for sine and cosine functions is:
\[ \sin(a + b) = \sin a \cos b + \cos a \sin b \]
and
\[ \cos(a + b) = \cos a \cos b - \sin a \sin b \.
\] Even though our focus here is on the tangent function, understanding these identities is important as they form the basis for the tan addition formula mentioned previously. Angle sum identities illustrate how trigonometric functions of an angle sum can be evaluated in terms of trigonometric functions of the individual angles themselves. This property is especially useful when analyzing periodic properties or symmetries within trigonometric expressions.

Trigonometric Identities

Trigonometric identities are equations that involve trigonometric functions and are true for all values of the included angles for which both sides of the identity are defined. Some basic trigonometric identities include the reciprocal identities such as \( \tan x = \frac{\sin x}{\cos x} \) and the Pythagorean identities like \( \sin^2 x + \cos^2 x = 1 \).

The product-to-sum and sum-to-product identities are extensions that demonstrate the relationships among trigonometric functions at various angle sums or differences. Mastery of these identities provides a robust foundation, allowing students to simplify and solve more complex trigonometric equations and proofs.

Mathematical Induction

Mathematical induction is a proof technique that is used to demonstrate the truth of an infinite sequence of statements. The process has two steps: proving a base case and then proving a general case.
1. Base Case: Show the statement holds for the initial value, usually \( n = 1 \).
2. Inductive Step: Assume the statement holds for \( n \), and then prove it holds for the next value, \( n + 1 \).

While the problem at hand does not directly involve a step-by-step induction process, breaking down a trigonometric problem into a product of expression allows the application of a similar logical process. By showing that pairs of tangent terms simplify to a known value, we effectively demonstrate that a pattern holds across the entire product, culminating in the final form of the expression.