Problem 85 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

$$ \cos 2 x-3 \cos x+1=\frac{1}{(\cot 2 x-\cot x) \sin (x-\pi)} $$

Step-by-Step Solution

Answer

Note that the given equation is difficult and may not have a general closed-form solution. The answer depends on the conditions of the problem, such as any restrictions on the values of \( x \). In reality, it would probably be solved numerically.

Step 1: Applying Trigonometric Identities

By using the double-angle identity \( \cos 2x = 1 - 2 \sin^2 x \), the equation becomes \( 1 - 2 \sin^2 x - 3 \cos x + 1 = \frac{1}{( \cot 2x - \cot x) \sin(x - \pi )} \)

Step 2: Incorporating Cotangent Identity

The cotangent function can be expressed in terms of sine and cosine functions as \( \cot x = \frac{\cos x}{\sin x } \). Thus, \( \cot 2x - \cot x \) becomes \( \frac{\cos 2x}{\sin 2x} - \frac{\cos x}{\sin x} \)

Step 3: Solving the Equation

Input the modified cotangent function back into the original equation and try to solve it, what part should equal 0, or what part should equal 1, etc, to make the whole equation true.

Key Concepts

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for every value of the occurring variables where both sides of the equation are defined. These identities are crucial for solving trigonometric equations, simplifying expressions, and evaluating integrals. Some commonly used trigonometric identities include:

• Pythagorean Identities, such as \( \sin^2 x + \cos^2 x = 1 \). This represents the fundamental relationship between sine and cosine functions.
• Reciprocal Identities, like \( \csc x = \frac{1}{\sin x} \) and \( \sec x = \frac{1}{\cos x} \).
• Quotient Identities, such as \( \tan x = \frac{\sin x}{\cos x} \) and \( \cot x = \frac{\cos x}{\sin x} \).

In the given exercise, the double angle identity \( \cos 2x = 1 - 2 \sin^2 x \) is used to transform the equation into a form that's easier to solve. Applying identities like these helps in viewing equations from a different angle, making it easier to identify solutions.

Double Angle Formulas

The double angle formulas are specific cases of angle formulas that relate the trigonometric functions of an angle to the trigonometric functions of double that angle. These can simplify the process of working with trigonometric expressions or solving equations. Here are some important double angle formulas:

• \( \cos 2x = \cos^2 x - \sin^2 x \)
• \( \cos 2x = 2 \cos^2 x - 1 \)
• \( \cos 2x = 1 - 2 \sin^2 x \)

For solving the given problem, the formula \( \cos 2x = 1 - 2 \sin^2 x \) was utilized. This formula is particularly useful when simplifying expressions or converting between sine and cosine. By replacing \( \cos 2x \) with \( 1 - 2 \sin^2 x \), the equation is simplified to help identify or isolate variables, leading to a clearer path to the solution.

Cotangent Function

The cotangent function, represented as \( \cot x \), is the reciprocal of the tangent function. It can be expressed as the ratio of cosine to sine: \( \cot x = \frac{\cos x}{\sin x} \). Understanding this function is key to solving trigonometric equations, especially those involving multiple angles.In the exercise, you are required to express \( \cot 2x - \cot x \) in terms of sine and cosine:

• \( \cot 2x = \frac{\cos 2x}{\sin 2x} \)
• \( \cot x = \frac{\cos x}{\sin x} \)

This manipulation is crucial to reformulate the original equation properly, considering the role of each trigonometric function involved. By re-expressing the problem in terms of sin and cos, we can more easily identify certain properties or simplifications that lead toward a solution. This approach helps in transforming the original complex expression into a simpler one that is more manageable to solve.