Problem 35 - Comprehensive Trigonometry for IIT JEE Main and Advanced Rejaul Makshud MeGraw Hill

Question

If \(\frac{\tan \theta}{\tan \theta-\tan 3 \theta}=\frac{1}{3}\), then find the value of \(\frac{\cot \theta}{\cot (\theta)-\cot (3 \theta)}\)

Step-by-Step Solution

Answer

The value of \( \frac{\cot \theta}{\cot (\theta)-\cot (3 \theta)} \) is \( \frac{1}{3} \).

Step 1: Recall the Relationship between Tangent and Cotangent

Before diving into the problem, remember the basic relationship between tangent and cotangent functions. We know that \( \cot \theta = \frac{1}{\tan \theta} \). This relationship will be used to express the two functions in terms of each other.

Step 2: Analyze the given Equation

We are given the equation \( \frac{\tan \theta}{\tan \theta-\tan 3 \theta} = \frac{1}{3} \). Isolating \( \tan \theta \) gives \( \tan \theta = \frac{\tan 3 \theta}{2} \).

Step 3: Convert Tangent to Cotangent

Using the inverse relationship between tangent and cotangent, we can express the equation obtained in step 2 using cotangent. \( \cot \theta = \frac{2}{\cot 3 \theta} \).

Step 4: Insert the value in Required Expression

Now, insert the value obtained in step 3 into the expression we want to find: \( \frac{\cot \theta}{\cot \theta-\cot 3 \theta} = \frac{\frac{2}{\cot 3 \theta}}{\frac{2}{\cot 3 \theta} - \cot 3 \theta} \). After simplifying, the result will be \( \frac{1}{3} \).

Key Concepts

Tangent and Cotangent Relationship

The relationship between the tangent and cotangent functions is a key concept in trigonometry. Understanding this relationship is crucial for solving many trigonometric equations and identities. The tangent of an angle, denoted as \( \tan \theta \), is the ratio of the opposite side to the adjacent side in a right-angled triangle. On the other hand, the cotangent of an angle, \( \cot \theta \), is the reciprocal of the tangent function. This means that:

• \( \cot \theta = \frac{1}{\tan \theta} \)
• Similarly, \( \tan \theta = \frac{1}{\cot \theta} \)

These reciprocal identities allow us to express tangent functions in terms of cotangent and vice versa, providing flexibility in solving trigonometric problems. This reciprocal relationship is often used to simplify expressions or transform them to a form that is more convenient for calculation, as demonstrated in the provided exercise and solution.

Simplifying Trigonometric Expressions

Simplifying trigonometric expressions often involves the substitution of equivalent trigonometric identities and relationships to reach a more manageable form. In mathematics, simplification is a crucial step as it reduces complexity and helps gain insights into the problem more easily.

• For example, using identities such as \( \tan \theta = \frac{\sin \theta}{\cos \theta} \) or \( \cot \theta = \frac{\cos \theta}{\sin \theta} \) can help express terms in ways that reveal cancellations or common factors.
• Transforming a tangent into a cotangent expression, or vice versa, can sometimes lead to simplifications by turning a division problem into multiplication.

In our context, by using the relationship \( \tan \theta = \frac{1}{\cot \theta} \), we transformed the equation from its original form into an equivalent expression with cotangent, aiding in the simplification process. Observing how these transformations simplify our work helps in gaining a deeper understanding of trigonometric manipulation.

Solving Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the variable. Solving these involves finding the values of the trigonometric function, or the angle, that make the equation true.

• In the exercise, we started with an identity involving the tangent function: \( \frac{\tan \theta}{\tan \theta - \tan 3\theta} = \frac{1}{3} \).
• By manipulating the equation, we isolated \( \tan \theta \) to find its relationship with \( \tan 3\theta \): \( \tan \theta = \frac{\tan 3\theta}{2} \).
• Subsequently, this expression was converted to involve cotangent, forming \( \cot \theta = \frac{2}{\cot 3\theta} \).

Solving such identities involves strategic rearranging and using known trigonometric relationships to transform complex expressions into simpler ones. Mastery in this process allows for resolving complex trigonometric problems efficiently and effectively. The solution showed how to substitute and simplify until the desired equation was reached, revealing a step-by-step method to handle these identities.