Problem 127 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

$$ \frac{1-\cos 2 A}{1+\cos 2 A}=\tan ^{2} A $$

Step-by-Step Solution

Answer

The original expression can be simplified and proven to be equal to \( \tan^{2}A \).

Step 1: Identify the Required Identities

First, recall the formula for cosine of double angle, that is \( \cos 2A = 1 - 2 \sin^{2}A \) or \( \cos 2A = 2 \cos^{2}A - 1 \). Also remember the formula for tangent, \( \tan A = \frac{\sin A}{\cos A} \). These identities will be useful in simplifying the equation.

Step 2: Apply the Identities

Substituting \( \cos 2A = 1 - 2 \sin^{2}A \) into the original equation gives \( \frac{1 - (1 - 2 \sin^{2}A)}{1 + (1 - 2 \sin^{2}A)} = \tan^{2}A \). Simplifying this further gives \( \frac{2 \sin^{2}A}{2 \cos^{2}A} = \tan^{2}A \)

Step 3: Final Simplification

The equation \( \frac{2 \sin^{2}A}{2 \cos^{2}A} = \tan^{2}A \) can be further simplified by dividing by 2 to give \( \frac{\sin^{2}A}{\cos^{2}A} = \tan^{2}A \). Since \( \tan A = \frac{\sin A}{\cos A} \), the left side of the equation can be rewritten to match the right side exactly, giving us \( \tan^{2}A = \tan^{2}A \)

Key Concepts

Double Angle Formulas

The Double Angle Formulas are essential tools in trigonometry used to simplify expressions or solve equations that involve trigonometric functions at double angles. These formulas express trigonometric functions like sine, cosine, and tangent of double angles in terms of single angle trigonometric functions. For cosine, two common forms are:

• \( \cos 2A = \cos^2 A - \sin^2 A \)
• Alternative forms: \( \cos 2A = 2\cos^2 A - 1 \) or \( \cos 2A = 1 - 2\sin^2 A \)

The choice among these forms depends on the other terms in your equation and what you're solving for. Here, we used \( \cos 2A = 1 - 2\sin^2 A \) to help establish an equation involving only sine functions, aiding in simplifying the given trigonometric identity. Understanding how to apply these formulas effectively is crucial in trigonometry, as they allow for the transformation of expressions which can lead to easier and more solvable equations.

Tangent Function

The Tangent Function is one of the foundational trigonometric functions. It is defined as the ratio of the sine and cosine of an angle. Specifically, for an angle \( A \), the tangent function is given by:

• \( \tan A = \frac{\sin A}{\cos A} \)

In the context of the exercise, the tangent function was used to simplify the expression \( \frac{1 - \cos 2A}{1 + \cos 2A} \). By substituting identities for \( \cos 2A \) and rearranging, the equation simplifies down to a pure tangent squared function \( \tan^2 A \). Recognizing and utilizing the relationship of tangent with other functions, like sine and cosine, can help deconstruct complex trigonometric expressions. It is especially useful for confirming identities, where matching one side of an equation to another is necessary.

Simplification Techniques

Simplification Techniques in trigonometry involve breaking down complex expressions into simpler parts that are easier to handle and understand. This often involves substituting identities like the double angle formulas and converting expressions from one type of trigonometric function to another.

• First, identify which trigonometric identities can simplify the expression.
• Use algebraic manipulation, such as factoring, expanding, or combining like terms, to reduce complexity.
• Substitute known identities, such as \( \cos 2A = 1 - 2\sin^2 A \), to transform the expression.
• Match transformed expressions to verify or solve identities, like turning \( \frac{\sin^2 A}{\cos^2 A} \) into \( \tan^2 A \).

These steps not only simplify solving trigonometric equations but also aid in proving identities. Approaching problems methodically with simplification techniques fosters a deeper understanding of trigonometry and enhances problem-solving efficiency.