Problem 300 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

$$ \text { If } \tan y=\frac{n \sin x \cos x}{1-n \sin ^{2} x}, \text { prove that } \tan (x-y)=(1-n) \tan x . $$

Step-by-Step Solution

Answer

By expressing \(tan y\) in terms of \(sin y\) and \(cos y\), then substituting them into the formula for \(tan (x - y)\), careful simplification confirms that indeed \(tan(x - y) = (1-n) \tan x\) as was required to be proved.

Step 1: Express \(\tan y\) in terms of \(\sin y\) and \(\cos y\)

By definition, \(\tan y = \frac{\sin y}{\cos y}\), and from the problem, it is given that \(\tan y= \frac{n \sin x \cos x}{1-n \sin ^{2} x}\), by equating the two and expressing \(\sin y\) and \(\cos y\) in terms of \(\sin x\) and \(\cos x\) we get \(\sin y = n \sin x \cos x\) and \(\cos y = 1-n \sin ^{2} x\)

Step 2: Use the formula for \(\tan(x - y)\)

Formula for \(\tan(x - y)\) is given by \(\frac{\tan x - \tan y}{1 + \tan x \tan y}\). Substitute the expressions for \(\tan x\) and \(\tan y\) from step 1 in the formula we get \(\tan(x - y)\) equals to \(\frac{\tan x - \frac{n \sin x \cos x}{1-n \sin ^{2} x}}{1 + \tan x \frac{n \sin x \cos x}{1-n \sin ^{2} x}}\)

Step 3: Simplify

On simplifying and performing the algebraic operations in step 2, we will achieve that \(\tan (x - y) = (1-n) \tan x\), which is exactly what was needed to be proved

Key Concepts

Tangent Function

The tangent function, often noted as \( \tan \theta \), is a fundamental trigonometric function. Its definition is the ratio of the sine and cosine functions. Specifically, \( \tan \theta = \frac{\sin \theta}{\cos \theta} \). This ratio represents how much the sine and cosine functions stretch or shrink relative to each other in a right triangle.

Tangent functions are periodic with a period of \( \pi \) radians. This means the function repeats its values every \( \pi \) units. It is undefined where \( \cos \theta = 0 \) since division by zero is not allowed. This occurs at odd multiples of \( \frac{\pi}{2} \) radians, known as vertical asymptotes in the function's graph.

Understanding \( \tan \theta \) is crucial as it frequently appears in various trigonometric identities and equations, helping to solve for angles and lengths in triangles. In this exercise, the tangent of an angle \( y \) is expressed in terms of another angle \( x \), illustrating how inter-dependent these functions can be in trigonometric identities.

Trigonometric Proofs

Trigonometric proofs involve verifying certain equations or expressions through known identities and properties of trigonometric functions. They often require creativity and a thorough understanding of trigonometric relationships.

To prove trigonometric identities, one commonly employs:

• Basic trigonometric identities, like \( \sin^2 \theta + \cos^2 \theta = 1 \).
• Angle sum and difference formulas, such as \( \tan(x-y) = \frac{\tan x - \tan y}{1 + \tan x \tan y} \).
• Reciprocal identities, like \( \tan \theta = \frac{1}{\cot \theta} \).

In the provided exercise, the task is to prove that \( \tan(x-y) = (1-n) \tan x \). This involves substituting the given expression for \( \tan y \) and simplifying using the difference formula for tangent. Understanding these proofs helps students engage with the logical reasoning required in mathematics, allowing them to manipulate and verify complex expressions.

Simplification of Trigonometric Expressions

Simplifying trigonometric expressions is an important skill in solving trigonometric equations and proofs. Simplification often involves reducing a complex expression into a simpler form using trigonometric identities and algebraic manipulations.

Key strategies include:

• Combining like terms by factoring trigonometric identities.
• Using substitution to replace one trigonometric function with another, based on known identities.
• Approximating expressions by cancelling terms where possible, thus simplifying the overall calculation.

In the step-by-step solution provided, simplification is used to achieve the final proof that \( \tan(x-y) = (1-n)\tan x \). The process illustrates how applying these strategies can connect different parts of a problem to form a cohesive and simpler solution.

Mastering this technique not only aids in solving trigonometric problems but also develops overall problem-solving abilities that are beneficial across various mathematical contexts.