Problem 24 - Comprehensive Trigonometry for IIT JEE Main and Advanced Rejaul Makshud MeGraw Hill

Question

If \(\theta=\frac{\pi}{2^{n}+1}\), then find the value of \(2^{n} \cos (\theta) \cos (2 \theta) \cos \left(2^{2} \theta\right) \ldots \cos \left(2^{n-1} \theta\right)\)

Step-by-Step Solution

Answer

The value of \(2^{n} \cos (\theta) \cos (2 \theta) \cos (2^{2} \theta) \ldots \cos (2^{n-1} \theta)\) is 0

Step 1: Simplify the expression using cosine multiplication

Here we can use multiplying factorials concept which states that \( \cos(\alpha) * \cos(2 \alpha) *…* \cos (2^{n-1} \alpha) = \frac{1}{2^n}* \sin(2^n}\alpha )\)

Step 2: Substitute given value of \(\theta\)

With \(\theta=\frac{\pi}{2^{n}+1}\), plug this value of \(\alpha\) into \(\frac{1}{2^n} \sin(2^n \alpha)\). \n Result is \(\frac{1}{2^n} * \sin(2^n *\frac{\pi}{2^{n}+1})\) = \(\frac{1}{2^n} * \sin(\pi)\)

Step 3: Calculate final value

Calculate \( \sin(\pi)\) which is 0. \n Therefore, the final output is \(0\).

Key Concepts

Cosine Multiplication

Cosine multiplication is a crucial concept in trigonometry, especially when dealing with product-to-sum identities or simplifying the product of multiple cosine terms. In many trigonometric expressions, you may encounter products like \( \text{cos}(\theta) \text{cos}(2\theta) \) that need to be simplified. The trigonometric identity for cosine multiplication helps us to transform the product of cosines into a sum, making it easier to simplify and solve. However, in certain cases, such as this exercise, a pattern emerges which allows us to apply a general formula.
Here, the multiplying factorials concept states that the product of cosines in a geometric progression, like \( \text{cos}(\theta) * \text{cos}(2\theta) * \text{cos}(2^2\theta) * \text{...} * \text{cos}(2^{n-1}\theta) \), where each term’s angle is a power of two times the previous term's angle, can be represented more compactly. This pattern leads to a powerful simplification, where the product of such cosines is equal to \( \frac{1}{2^n} * \text{sin}(2^n \theta) \).
Using this identity, we can greatly reduce the complexity of our given trigonometric expressions, allowing for simpler and more accessible solutions.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the occurring variables where both sides of the equation are defined. These identities are invaluable for simplifying and transforming trigonometric expressions and for solving trigonometric equations. Common basic trigonometric identities include the Pythagorean identities, angle sum and difference identities, double angle formulas, and the half-angle formulas.
In the context of the given exercise, we mainly employ the simplification techniques provided by trigonometric identities. The identity for multiplying cosines highlights how interconnected and versatile these identities are. Furthermore, knowing that \( \text{sin}(\theta) = 0 \) when \( \theta \) is an integer multiple of \( \text{pi} \) is another trigonometric identity that directly aids in solving the final step of the problem. Mastery of these identities is not only crucial for success in IIT JEE trigonometry problems but for understanding the subject of trigonometry as a whole.

Angle Multiplication in Trigonometry

In trigonometry, angle multiplication involves finding the trigonometric functions of multiples of an angle, such as double, triple, or any integer multiple of the angle. The angle multiplication formulas let us express functions like \( \text{sin}(2\theta) \), \( \text{cos}(2\theta) \), or even \( \text{sin}(n\theta) \) and \( \text{cos}(n\theta) \) in terms of trigonometric functions of the original angle \( \theta \). These formulas are essential when you need to integrate, differentiate, or simplify trigonometric expressions.
The exercise given leverages a specific case of angle multiplication where the successive multiplication involves angles that are powers of two. The solution uses a special case formula that significantly simplifies the calculation by converting the product of cosines into a single sine function of the multiplied angle. Being proficient in recognizing patterns like angle multiplication sequences can drastically streamline solving complex trigonometric problems, as seen in the step-by-step solution of our original problem.