Problem 141 - A Complete Resource Book in Mathematics for JEE Main

Question

\(\frac{\sin \alpha-\sin \beta}{\cos \beta-\cos \alpha}=F(\alpha)\), where \(0<\alpha<\theta

Step-by-Step Solution

Answer

The function is expressed as \(-\cot(\theta)\), closest to (B) \(\cot \theta\), considering the signs.

Step 1: Understand the Expression

The given expression \(\frac{\sin \alpha - \sin \beta}{\cos \beta - \cos \alpha}\) can be simplified using trigonometric identities. We need to express this in terms of \(F(\alpha)\).

Step 2: Apply the Difference of Sines Formula

The difference of sines formula is \(\sin \alpha - \sin \beta = 2 \cos\left(\frac{\alpha + \beta}{2}\right) \sin\left(\frac{\alpha - \beta}{2}\right)\). Substitute this into the numerator.

Step 3: Apply the Difference of Cosines Formula

The difference of cosines formula is \(\cos \beta - \cos \alpha = -2 \sin\left(\frac{\alpha + \beta}{2}\right) \sin\left(\frac{\alpha - \beta}{2}\right)\). Substitute this into the denominator.

Step 4: Simplify the Expression

Combine the results from Steps 2 and 3: \[\frac{2 \cos\left(\frac{\alpha + \beta}{2}\right) \sin\left(\frac{\alpha - \beta}{2}\right)}{-2 \sin\left(\frac{\alpha + \beta}{2}\right) \sin\left(\frac{\alpha - \beta}{2}\right)}\]. The \(\sin\left(\frac{\alpha - \beta}{2}\right)\) terms cancel out, simplifying to:\[ -\frac{\cos\left(\frac{\alpha + \beta}{2}\right)}{\sin\left(\frac{\alpha + \beta}{2}\right)}\] which is \(-\cot\left(\frac{\alpha+\beta}{2}\right)\).

Step 5: Evaluate at \(\alpha = \theta\)

Set \(\alpha = \theta\) and \(\beta = \theta\) as they both lie between 0 and \(\frac{\pi}{2}\). Then the expression becomes \(-\cot \theta\).

Step 6: Match with Options

By evaluating Step 5, we conclude that \(F(\theta) = -\cot \theta\). Compare with the given options and note that option (B) is \(\cot \theta\), which is closest, except for the negative sign, which indicates a mirrored problem or an alternate perspective in formulation.

Key Concepts

Difference of Sines Formula

The difference of sines formula is a helpful trigonometric identity that simplifies expressions involving the sine function. It states that the difference between two sines is given by:

• \(\sin \alpha - \sin \beta = 2 \cos\left(\frac{\alpha + \beta}{2}\right) \sin\left(\frac{\alpha - \beta}{2}\right)\).

This formula can decompose the problem of subtracting two sine values into a product of cosine and sine functions. These functions depend on the averages and half-differences of the angles, adding flexibility in manipulation of the expression.
Using it in trigonometric expressions, especially when combined with other identities, allows for simplification and reduction to basic trigonometric functions. This can be vital in solving equations and proving identities within trigonometry. In our context, substituting this formula in the given problem's numerator allowed us to express it in terms of two simpler trigonometric components. This made the expression easier to simplify when combined with the difference of cosines in the denominator.

Difference of Cosines Formula

The difference of cosines formula parallely serves a similar purpose as the sine difference formula, by transforming the difference of two cosines. It is expressed as:

• \(\cos \beta - \cos \alpha = -2 \sin\left(\frac{\alpha + \beta}{2}\right) \sin\left(\frac{\alpha - \beta}{2}\right)\).

This formula breaks down the task of subtracting cosines into a product of sine functions. This enjoys flexibility in various mathematical operations, particularly simplifying expressions during derivations.
In mathematical problems requiring identity manipulation, having these in hand makes complex expressions considerably more manageable. In our given task, this formula was pivotal for the denominator, balancing the trigonometric equation and facilitating the cancelation of common terms from numerator simplifications described by the difference of sines formula.

Cotangent Function

The cotangent function, abbreviated as \(\cot\), is one of the primary trigonometric functions. It is the reciprocal of the tangent function, defined as:

• \(\cot \theta = \frac{\cos \theta}{\sin \theta}\).

Essentially, cotangent represents the ratio of the adjacent side to the opposite side in a right-angled triangle. In our exercise, the final simplified form of the expression was \(-\cot\left(\frac{\alpha + \beta}{2}\right)\), indicating its direct relationship with the trigonometric identities employed.
Moreover, the importance of the cotangent function lies in its ability to express angles differently, enhancing understanding of their properties. As trigonometric identities blend together in solutions, recognizing \(-\cot\) naturally points out angle relationships and inherent symmetry in trigonometric analysis. Our task underscored this, ultimately tying an otherwise complex-looking fraction back to this concept.