Problem 186 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

$$ \frac{\sin (n+1) A+2 \sin n A+\sin (n-1) A}{\cos (n-1) A-\cos (n+1) A}=\cot \frac{A}{2} $$

Step-by-Step Solution

Answer

The given trigonometric expression does indeed simplify to \(\cot \frac{A}{2}\). The crucial steps are the application of the sine and cosine of sum and difference identities, and the simplification of the resulting function.

Step 1: Simplifying the Trigonometric Functions

Starting off with the given equation: \[ \frac{\sin (n+1) A+2 \sin n A+\sin (n-1) A}{\cos (n-1) A-\cos (n+1) A}=\cot \frac{A}{2} \] Both the numerator and the denominator can be simplified using the concept of sine of sum and difference of two numbers: \[ \sin(A+B) = \sin A \cos B + \cos A \sin B \] and \[ \cos(A-B) = \cos A \cos B + \sin A \sin B \] So applying these rules, we have: \[ = \frac{\sin nA \cos A + \cos nA \sin A + 2 \sin nA + \sin nA \cos A - \cos nA \sin A}{\cos nA \cos A + \sin nA \sin A - \cos nA \cos A + \sin nA \sin A} \]

Step 2: Cancelling the Terms

On simplification, terms cancel each other in numerator and denominator: \[ = \frac{2 \sin nA \cos A + 2 \sin nA}{2 \sin nA \sin A} \] Simplify further: \[ = \frac{\sin nA (2 \cos A + 2)}{\sin nA (2 \sin A)} \] The \(\sin nA\) will then cancel out: \[ = \frac{2 (\cos A + 1)}{2 \sin A} \]

Step 3: Using Cotangent

It can be simplified to: \[ = \cot \frac{A}{2} = \frac{2 (\cos A + 1)}{\sin A} \] This is the simplified version of the given expression and it equals to \(\cot \frac{A}{2}\), which states that the two sides of the equation are equal and the identity presented is confirmed as being correct.

Key Concepts

Sine of Sum and Difference

Understanding the sine of sum and difference is pivotal when dealing with trigonometric expressions. This concept is founded on the identities that express the sine of the sum or difference of two angles in terms of the products of sines and cosines of these angles. For instance, the identity
\[ \sin(A + B) = \sin A \cos B + \cos A \sin B \]
expresses the sine of the sum of two angles, A and B. In contrast,
\[ \sin(A - B) = \sin A \cos B - \cos A \sin B \]
represents the sine of the difference. These identities are instrumental for simplifying trigonometric expressions and proving other identities. For example, if we look at the given problem, we can use these identities to break down the more complex terms involving \( \sin(n+1)A \) and \( \sin(n-1)A \) into expressions involving \( \sin nA \) and \( \cos nA \), which then allows further simplification.

Cotangent Simplification

Cotangent is one of the six fundamental trigonometric functions, and simplifying expressions that involve cotangent often requires other trigonometric identities. The cotangent of an angle, which is the reciprocal of the tangent, can also be expressed as the ratio of the cosine to the sine of that angle, as shown:
\[ \cot A = \frac{\cos A}{\sin A} \]
In the context of the provided exercise, simplification reaches a point where the expression becomes
\[ \frac{2 (\cos A + 1)}{2 \sin A} \]
This can be seen as \( \cot \frac{A}{2} \) due to what's known as the half-angle identity, which makes a bridge between the expression and cotangent function:
\[ \cot \frac{A}{2} = \frac{\cos \frac{A}{2}}{\sin \frac{A}{2}} \]
These identities streamline the process of simplifying trigonometric expressions significantly, especially when dealing with complex problems or proofs, and hence, are fundamental to mastering trigonometry.

Trigonometry for IIT JEE

Trigonometry is a profound and essential topic for the IIT JEE, one of the most competitive engineering entrance exams in India. Proficiency in trigonometric identities, including the sine of sum and difference, cotangent simplification, and numerous others, is necessary for success.
For students preparing for IIT JEE, mastering these identities allows for efficient problem-solving and the ability to tackle a variety of questions. It's not just about memorizing formulas; candidates need to understand the underlying concepts to apply them flexibly.
From solving equations to proving identities, trigonometry questions in IIT JEE often challenge students to think critically and creatively. As seen in the exercise provided, the journey of simplification through the application of these identities is a classic illustration of IIT JEE-level problem-solving. Hence, investing time to practice and grasp the principles of trigonometry can significantly aid in aspiring engineers' exam preparations.