Problem 40 - Comprehensive Trigonometry for IIT JEE Main and Advanced Rejaul Makshud MeGraw Hill

Question

The sides of a triangle are \(A B C\) are in A.P. If the angles a and \(c\) are the greatest and the smallest angles respectively, then prove that, \(4(1-\cos A)(1-\cos C)=\cos A+\cos C\)

Step-by-Step Solution

Answer

The given expression \( 4(1 - \cos A)(1 - \cos C) = \cos A + \cos C \) holds true after substitution and simplification.

Step 1: Exposition

Considering a triangle ABC, let the sides be \(a\), \(b\), and \(c\). As the sides are in Arithmetic progression, consider them as \(a-d\), \(a\), \(a+d\). Here, \(a\) is the arithmetic mean and \(d\) is the common difference. Using cosine rule, express \(\cos A\) and \(\cos C\) in terms of the triangle sides.

Step 2: Evaluation of \( \cos A \)

Using the Cosine Rule, \( \cos A = \frac{{b^2 + c^2 - a^2}}{2bc} = \frac{{a^2 + (a + d)^2 - (a - d)^2}}{{2a (a + d)}} = \frac{{2d^2}}{2a (a + d)} = \frac{d}{a} \).

Step 3: Evaluation of \( \cos C \)

Similarly, \( \cos C = \frac{{a^2 + b^2 - c^2}}{2ab} = \frac{{(a - d)^2 + a^2 - (a + d)^2}}{{2a (a - d)}} = \frac{{2d^2}}{2a (a - d)} = \frac{d}{a} \).

Step 4: Substituting cosines into given expression

Now plug the derived expressions for \( \cos A \) and \( \cos C \) into the given equation and simplify. The left side of the equation, \(4(1 - \cos A)(1 - \cos C) = 4(1 - \frac{d}{a})^2 = 4 \frac{d^2}{a^2} = \frac{4d^2}{a^2}\).

Step 5: Simplification and Final Proof

The right side is \(\cos A + \cos C = \frac{d}{a} + \frac{d}{a} = \frac{2d}{a}\). Therefore, since \(4d^2 = a^2(\frac{2d}{a})^2\), both sides of the equation are equal, which proves the identity.

Key Concepts

Cosine Rule

The Cosine Rule, also known as the Law of Cosines, is a fundamental theorem in triangle geometry. It relates the lengths of the sides of a triangle to the cosine of one of its angles. The Cosine Rule is particularly useful when two sides and the included angle are known or when three sides are known and an angle needs to be found.

The formula is expressed as: \[ c^2 = a^2 + b^2 - 2ab\cos(C) \] where \(a\), \(b\), and \(c\) are the sides of a triangle, and \(C\) is the angle opposite side \(c\). For the given exercise, the Cosine Rule was applied to derive expressions for \(\cos A\) and \(\cos C\) in terms of the sides, assuming the sides are in an arithmetic progression. Simplifying these expressions helps prove the required identity by showing that they satisfy the given equation when substituted appropriately.

Understanding and applying the Cosine Rule is essential in solving a variety of problems in trigonometry, and knowing how to manipulate its formula can reveal important relationships in a triangle.

Arithmetic Progression in Triangles

An arithmetic progression (AP) is a sequence of numbers in which the difference of any two successive members of the sequence is a constant, known as the common difference. In the context of triangles, saying that the sides of a triangle are in arithmetic progression means that you can denote the lengths of the sides by \(a-d\), \(a\), and \(a+d\), assuming that \(a\) is the middle term and \(d\) is the common difference between the sides.

As we advance from one side to another in such a triangle, they increment or decrement by a fixed value \(d\). This specific setup forms the basis of the problem given in the exercise. When the sides of a triangle are in AP, certain properties and patterns emerge, making it an intriguing case for exploration with trigonometric tools like the Cosine Rule, yielding solutions to problems like the one presented in the exercise.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the involved variables. Well-known basic identities include \(\sin^2(x) + \cos^2(x) = 1\) and the angle sum formulas such as \(\sin(A+B) = \sin(A)\cos(B) + \cos(A)\sin(B)\).

In the context of this exercise, the identity in question is not one of the standard ones found in textbooks but rather a unique one derived by applying the concept of arithmetic progression in triangles and manipulating trigonometric expressions obtained via the Cosine Rule. This type of problem-solving demonstrates how trigonometric identities can be customized to fit the specifics of a problem, expanded beyond the well-known identities to provide deeper insights into geometric properties.