Problem 272 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

$$ \text { If } m \cos (\theta+\alpha)=n \cos (\theta-\alpha), \text { show that }(m-n) \cot \theta=(m+n) \tan \alpha \text { . } $$

Step-by-Step Solution

Answer

The short answer to the exercise after manipulations and simplification is \( (m-n) \cot \theta = (m+n) \tan \alpha \).

Step 1: Conversion of Cosine Identities

First, convert \( \cos(\theta + \alpha) \) and \( \cos(\theta - \alpha) \) into cosine subtraction and addition identities. Thus, \( m \cos(\theta +\alpha) \) becomes \( m [\cos \theta \cos \alpha - \sin \theta \sin \alpha] \) and \( n \cos(\theta - \alpha) \) turns into \( n [\cos \theta \cos \alpha + \sin \theta \sin \alpha] \).

Step 2: Equating converted identities

Setting the newly obtained forms of cosine functions equal to each other, we get:\( m [\cos \theta \cos \alpha - \sin \theta \sin \alpha]=n [\cos \theta \cos \alpha + \sin \theta \sin \alpha] \).

Step 3: Simplifying the equation

The above equation simplifies to \( (m-n) \cos \theta \cos \alpha = (m+n) \sin \theta \sin \alpha \). Divide both sides by \( \cos \theta \cos \alpha \sin \theta \sin \alpha \) to get \( (m-n) \cot \theta = (m+n) \tan \alpha \).

Key Concepts

Cosine Rule

The Cosine Rule, often referred to in trigonometry, is a vital tool for solving triangles when you don't have a right angle. While it's primarily known for its use in calculating the sides and angles of a triangle, this principle can be extended into various trigonometric identities. If you have a triangle with sides \(a\), \(b\), \(c\) and the angle \(C\) opposite to side \(c\), the Cosine Rule is expressed as:

• \(c^2 = a^2 + b^2 - 2ab \cos(C)\)

This equation connects the lengths of a triangle's sides with the cosine of one of its angles. When it's not possible to use simpler methods like the Pythagorean theorem due to the absence of a right angle, the Cosine Rule becomes extremely handy in solving for unknown angles or sides in any type of triangle.
Remember, it's not limited just to side length calculations. In some exercises, like this one, understanding how cosine interacts with angles can help you simplify expressions involving angle addition or subtraction.

Angle Addition and Subtraction

Understanding angle addition and subtraction is essential in trigonometry when working with complex angle expressions. The identities for these operations tell us how to express \( \cos(\theta + \alpha) \) and \( \cos(\theta - \alpha) \):

• \(\cos(\theta + \alpha) = \cos \theta \cos \alpha - \sin \theta \sin \alpha\)
• \(\cos(\theta - \alpha) = \cos \theta \cos \alpha + \sin \theta \sin \alpha\)

These equations break down an expression of cosine with a compound angle into more manageable parts.
This is particularly useful in exercises and equations where these angles appear, as it allows you to separate and manipulate them into simpler terms.
Knowing these identities is crucial for deriving other equations and for tasks like proving identities or finding unknown values in trigonometric contexts.

Trigonometry Equations

Trigonometric equations are equations that involve trigonometric functions. Solving them requires an understanding of various identities and sometimes a bit of algebraic manipulation.
For instance, when dealing with the given expression \(m \cos(\theta + \alpha) = n \cos(\theta - \alpha)\), you need to use trigonometric identities like the angle addition and subtraction formulas.
The next step involves setting both sides equal and simplifying them to uncover relationships such as \((m-n) \cot \theta = (m+n) \tan \alpha\).

• Begin by substituting using known identities - break complex terms into easier pieces.
• Use algebraic manipulation - move terms around to isolate the variable or expression of interest.
• Simplify further, possibly involving division or factoring, to express the desired relation.

Practice and familiarity with these equations and identities will help streamline your process in not just solving, but also in understanding the underlying principles governing these relationships.