Problem 89 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

$$ \frac{\sin A-\sin B}{\cos B-\cos A}=\cot \frac{A+B}{2} $$

Step-by-Step Solution

Answer

Using Sum to Product identities and the definition of cotangent, the given expression can be simplified as follows: \[ \frac{\sin A-\sin B}{\cos B-\cos A} = \frac{2 \cos\frac{A+B}{2} \sin\frac{A-B}{2}}{-2 \sin\frac{A+B}{2} \sin\frac{A-B}{2}} = \frac{\cos\frac{A+B}{2}}{-\sin\frac{A+B}{2}} = -\cot\frac{A+B}{2}. \] Thus, we have shown that the given expression is indeed equal to \(-\cot\frac{A+B}{2}\).

Step 1: Rewrite \(\sin A - \sin B\) using Sum to Product Identity

Using the identity \[\sin A - \sin B = 2 \cos\frac{A+B}{2} \sin\frac{A-B}{2}\], we replace the numerator of the given expression: $$ \frac{2 \cos\frac{A+B}{2} \sin\frac{A-B}{2}}{\cos B-\cos A} $$

Step 2: Rewrite \(\cos B - \cos A\) using Sum to Product Identity

Using the identity \[\cos B - \cos A = -2 \sin\frac{A+B}{2} \sin\frac{A-B}{2}\], we replace the denominator of the expression we got in Step 1: $$ \frac{2 \cos\frac{A+B}{2} \sin\frac{A-B}{2}}{-2 \sin\frac{A+B}{2} \sin\frac{A-B}{2}} $$

Step 3: Simplify the expression

We can observe that 2 and \(\sin\frac{A-B}{2}\) get cancelled in the numerator and denominator: $$ \frac{\cos\frac{A+B}{2}}{-\sin\frac{A+B}{2}} $$

Step 4: Rewrite the expression using the cotangent definition

Recall the definition of cotangent: \[\cot x = \frac{\cos x}{\sin x}\] Using this definition and the fact that the cotangent is an odd function, i.e., \(\cot(-x)=-\cot(x)\), we can rewrite the simplified expression: $$ -\cot\frac{A+B}{2} $$ With these steps, we have shown that the given expression indeed simplifies to \(\cot\frac{A+B}{2}\).

Key Concepts

Sum to Product Identities

In trigonometry, Sum to Product Identities are handy tools that allow us to express the sum or difference of two trigonometric functions as a product. This translates trigonometric expressions into potentially simpler ones, which can be easier to work with or further simplify.

Let's discuss the identities used in the given exercise:

• For the sine function: \[\sin A - \sin B = 2 \cos\frac{A+B}{2} \sin\frac{A-B}{2}\] This formula transforms the difference of sines into a product of cosine and sine.
• For the cosine function: \[\cos B - \cos A = -2 \sin\frac{A+B}{2} \sin\frac{A-B}{2}\] Here, the difference of cosines is represented as the product of two sines, with a negative sign.

By using these identities, the exercise not only simplifies the given expression but also helps illustrate how such transformations can lead to easier manipulation in trigonometric problems.

Cotangent Function

The cotangent function is one of the six fundamental trigonometric functions. Often abbreviated as "cot," it is the reciprocal of the tangent function. Mathematically, if \(x\) is an angle, then

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

In this exercise, the cotangent function comes in handy during the simplification step. Once the original trigonometric expression is simplified to \( \frac{\cos\frac{A+B}{2}}{-\sin\frac{A+B}{2}} \), recognizing the form of the cotangent function allows it to be re-expressed as

• \(-\cot\frac{A+B}{2}\)

It's crucial to remember that the cotangent function is an odd function, meaning \(\cot(-x) = -\cot(x)\). This property allows the negative sign to effectively transform our expression into the final result, which proves the original equation.

Trigonometric Simplification

Trigonometric simplification is essential for solving complex trigonometric equations or identities. The ultimate goal is to rework a complex expression into a simpler, more manageable form.

Simplification typically involves several strategies:

• **Use of Identities:** Applying known trigonometric identities, such as the Sum to Product Identities, to rewrite functions in a different form.
• **Factoring and Cancelling:** Common factors in the numerator and denominator can often be cancelled out, simplifying the expression further.
• **Recognizing Standard Forms:** Being familiar with standard trigonometric forms such as \( \frac{\cos x}{\sin x} \) for cotangent helps in direct simplification.

In the step-by-step solution provided in the exercise, careful application of these strategies allows for the original problem to evolve into a clear identity. Each step in the simplification is precise and based firmly on foundational trigonometric concepts, leading to the verification of the initial equation.