Problem 44 - Comprehensive Trigonometry for IIT JEE Main and Advanced Rejaul Makshud MeGraw Hill

Question

If \(\tan x=a\), then the value of \(\cot \left(\frac{\pi}{4}-a\right)\) is (a) \(\left(\frac{a-1}{a+1}\right)\) (b) \(\left(\frac{a^{2}-1}{a^{2}+1}\right)\) (c) \(\left(\frac{a^{2}+1}{a^{2}-1}\right)\) (d) \(\left(\frac{a+1}{a-1}\right)\)

Step-by-Step Solution

Answer

The value of \( \cot(\frac{\pi}{4} - a) \) is (d) \( \frac{a + 1}{a - 1} \).

Step 1: Understanding Tangent and Cotangent

The cotangent of an angle is the reciprocal of the tangent of that angle. Therefore, \( \cot x = \frac{1}{\tan x} = \frac{1}{a} \). Hence, we have \( a = \tan x = \frac{1}{\cot x} \).

Step 2: Finding relationship between \( \tan x \) and \( \cot (\frac{\pi}{4} - x) \)

Using the formula \( \cot(\frac{\pi}{4} - x) = \frac{1 + \tan x}{1 - \tan x} \), substitute \( a \) for \( \tan x \). This results in \( \cot(\frac{\pi}{4} - a) = \frac{1 + a}{1 - a} \).

Step 3: Comparing the results with the options

Comparing the obtained value with the available options, we see that \( \cot(\frac{\pi}{4} - a) = \frac{1 + a}{1 - a} \) is the same as option (d).

Key Concepts

Understanding Tangent

Tangent is one of the fundamental trigonometric functions used to relate the angles and sides of a right triangle. It is defined as the ratio of the opposite side to the adjacent side of an angle in a right triangle. Mathematically, this can be expressed as \( \tan \theta = \frac{\text{opposite}}{\text{adjacent}} \). However, tangent can also be understood in terms of the unit circle. In this context, it is described as the ratio of the y-coordinate to the x-coordinate of the point where the terminal side of an angle intersects the unit circle.

This fundamental definition of tangent helps in solving various trigonometric problems, particularly when converting between different forms of trigonometric expressions, as we are dealing with in this exercise. It allows for simplifications using identities or transformations.

Exploring Cotangent

Cotangent, often denoted as \( \cot \theta \), is closely related to the tangent function. It is defined as the reciprocal of the tangent. Therefore, \( \cot \theta = \frac{1}{\tan \theta} = \frac{\text{adjacent}}{\text{opposite}} \).

Like tangent, cotangent can also be described in the unit circle context as the ratio of the x-coordinate to the y-coordinate. Understanding cotangent as the reciprocal of tangent is essential because it allows conversions and simplifications between different trigonometric expressions, enabling quicker calculations. In our exercise, recognizing that \( \cot(\frac{\pi}{4} - x) \) involves transformation rules showcases the importance of this reciprocal relationship.

Pythagorean Identities

The Pythagorean identities are a group of fundamental relationships in trigonometry based on the Pythagorean theorem. They provide powerful tools for simplifying trigonometric expressions and solving equations.

• \( \sin^2 \theta + \cos^2 \theta = 1 \)
• \( 1 + \tan^2 \theta = \sec^2 \theta \)
• \( 1 + \cot^2 \theta = \csc^2 \theta \)

Each of these identities corresponds to different trigonometric functions and allows conversion from one form to another, depending on the need of the problem.

This is not directly applied in our current problem, but a thorough understanding of these identities prepares us for more complex trigonometric manipulations involving simplifications and transformations.

Angle Subtraction Formula

The angle subtraction formula is a critical identity in trigonometry used to find the tangent, cotangent, sine, or cosine of the difference of two angles. In our exercise, the focus is on the cotangent subtraction formula:
\[ \cot(\alpha - \beta) = \frac{1 + \tan \alpha \tan \beta}{\tan \beta - \tan \alpha} \]
Knowing this formula enables us to find the value of cotangent difference of two angles efficiently. This formula expresses how cotangent changes with angle subtraction and when applied, allows substitution into simpler forms.

For the given problem, substituting \( \tan x \) into \( \cot(\frac{\pi}{4} - a) \), we use this identity to derive the solution: \( \frac{1 + a}{1 - a} \). It's a vital tool in trigonometric calculations, essential for transformations and problem-solving.