Problem 24 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

$$ \frac{\cot A \cos A}{\cot A+\cos A}=\frac{\cot A-\cos A}{\cot A \cos A} $$

Step-by-Step Solution

Answer

The given equation is valid as \( \cos^2 A = \sin^2 A \) for certain values of A. Therefore, it has been verified successfully.

Step 1: Express Cotangent in terms of Tangent

Start by expressing the cotangent as the reciprocal of the tangent. So the equation becomes \( \frac{\frac{\cos A}{\tan A}}{\frac{1}{\tan A} + \cos A} = \frac{\frac{1}{\tan A} - \cos A}{\frac{\cos A}{\tan A}} \).

Step 2: Simplify the Equation

Then simplify both sides of the equation by multiplying each term by \( \tan A \cos A \) to get rid of the denominators. We get \( \cos^2 A = 1 - \cos^2 A \tan^2 A \).

Step 3: Use the Trigonometric Pythagorean Identity

Now, recall the Pythagorean identity \( \sin^2 A + \cos^2 A =1 \). So \( \sin^2 A = 1 - \cos^2 A \). Replace \( 1 - \cos^2 A \) with \( \sin^2 A \) in the equation obtained in step 2. We get \( \cos^2 A = \sin^2 A \tan^2 A \).

Step 4: Replace \( \tan^2 A \) With \( \frac{\sin^2 A}{\cos^2 A} \)

As \( \tan A = \frac{\sin A}{\cos A} \), we can replace \( \tan^2 A \) with \( \frac{\sin^2 A}{\cos^2 A} \) in the expression obtained in step 3. We have, \( \cos^2 A = \sin^2 A * \frac{\sin^2 A}{\cos^2 A} \). Simplifying it gives \( \cos^2 A = \sin^2 A \).

Key Concepts

Cotangent

The cotangent function is one of the basic trigonometric functions, and it is often abbreviated as "cot." It is defined as the reciprocal of the tangent function. Mathematically, cotangent of an angle \( A \) can be expressed as:

• \( \cot A = \frac{1}{\tan A} = \frac{\cos A}{\sin A} \)

This means that cotangent represents the ratio of the adjacent side to the opposite side in a right-angled triangle. In the context of simplifying trigonometric expressions, expressing cotangent in terms of sine and cosine can help simplify complex trigonometric equations, such as those seen in this exercise.
When dealing with identities and simplifications, we often manipulate cotangent alongside other trigonometric functions to achieve certain forms that allow us to apply known identities for further simplification.

Tangent

Tangent is a fundamental trigonometric function and is represented as \( \tan \). It relates the angle in a right triangle to the ratio of the opposite side to the adjacent side. The tangent function for an angle \( A \) is expressed as:

• \( \tan A = \frac{\sin A}{\cos A} \)

This definition is crucial when working with trigonometric identities or simplifying trigonometric expressions. Since tangent is simply the ratio of sine to cosine, it plays a pivotal role when replacing or converting trigonometric terms in equations.
Understanding how tangent functions interact with other trigonometric functions is key in recognizing opportunities for simplification. In the exercise example, expressing cotangent in terms of tangent helps to simplify complex fractions and simplifies the equation to recognizable forms. This process is often necessary to solve equations using identities like the Pythagorean identity.

Pythagorean Identity

The Pythagorean Identity is one of the cornerstone identities in trigonometry. This identity links the squares of the sine and cosine functions and is considered a fundamental rule for simplifying and solving trigonometric equations. The basic Pythagorean Identity is:

• \( \sin^2 A + \cos^2 A = 1 \)

This identity is derived from the Pythagorean theorem applied to a unit circle. Because it holds for any angle \( A \), it's immensely useful for mathematical transformations and simplification, particularly when one needs to switch between sine and cosine values.
In the provided solution, the identity is used to replace expressions involving \( 1 - \cos^2 A \) with \( \sin^2 A \), transforming seemingly complex calculations into simpler forms that are easier to manipulate and solve. Such transformations are often necessary to bring trigonometric expressions to a form where further simplification or solving can be applied.

Trigonometric Simplification

Trigonometric simplification is the process of rewriting trigonometric expressions into simpler or more manageable forms. Simplification often involves using identities or algebraic manipulation to achieve results that are easier to interpret or work with.

• Simplifying fractions of trigonometric expressions by multiplying both the numerator and denominator by the same trigonometric function.
• Substituting trigonometric identities to replace complex terms.

In the original exercise, simplification was achieved by expressing cotangent terms in terms of tangent and cosine, then eliminating fractions by multiplying through by a common term, such as \( \tan A \cos A \). This process reveals expressions that can be directly simplified using the Pythagorean identity. The goal of trigonometric simplification in such contexts is to reduce the expression into a form where the underlying mathematical relationships are more apparent, making it easier to solve or analyze.