Problem 135 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

$$ \tan \left(\frac{\pi}{4}+\theta\right)-\tan \left(\frac{\pi}{4}-\theta\right)=2 \tan 2 \theta . $$

Step-by-Step Solution

Answer

The original equation \( \tan \left(\frac{\pi}{4}+\theta\right)-\tan \left(\frac{\pi}{4}-\theta\right)=2 \tan 2 \theta \) holds true based on the transformation and simplification steps. No specific values of \( \theta \) are needed.

Step 1: Use the identities for the sum and difference of two angles

The first operation is to transform the tangent of the sum and difference of two angles on the left side of the equation to their identities, using the following formula: \( \tan (\alpha + \beta) = \frac{\tan \alpha + \tan \beta}{1 - \tan \alpha \tan \beta} \) and \( \tan (\alpha - \beta) = \frac{\tan \alpha - \tan \beta}{1 + \tan \alpha \tan \beta} \). When \( \alpha = \theta \) and \( \beta = \frac{\pi}{4} \), we get \( \tan \left(\frac{\pi}{4}+\theta\right) = \frac{1 + \tan \theta}{1 - \tan \theta} \) and \( \tan \left(\frac{\pi}{4}-\theta\right) = \frac{1 - \tan \theta}{1 + \tan \theta} \).

Step 2: Subtract the two results

Next, subtraction between these two results should be done, according to the original problem. The resulting equation is \( \frac{1 + \tan \theta}{1 - \tan \theta} - \frac{1 - \tan \theta}{1 + \tan \theta} = 2 \tan 2 \theta \). Simplify this equation by combining the two fractions to obtain \( \frac{2 \tan \theta}{1 - \tan^2 \theta} = 2 \tan 2 \theta \).

Step 3: Compare the left and right side

Observing both sides of the equation, the left side resembles the formula of \( \tan 2 \alpha = \frac{2 \tan \alpha}{1 - \tan^2 \alpha} \), where \( \alpha = \theta \). Thus, the left side essentially equals \( \tan 2 \theta \)

Key Concepts

Angle Sum and Difference Identities

The Angle Sum and Difference Identities are essential tools in trigonometry, especially when you need to simplify or solve trigonometric equations. These identities help to express the trigonometric functions of sums or differences of angles in terms of products, sums, or differences of trigonometric functions of individual angles.

For example, the formula for tangent sum identity is given by:

• \( \tan(\alpha + \beta) = \frac{\tan \alpha + \tan \beta}{1 - \tan \alpha \tan \beta} \)

Similarly, for the difference of angles, the identity is:

• \( \tan(\alpha - \beta) = \frac{\tan \alpha - \tan \beta}{1 + \tan \alpha \tan \beta} \)

When working with problems like \( \tan \left(\frac{\pi}{4}+\theta\right)-\tan \left(\frac{\pi}{4}-\theta\right) \), apply these identities to express the tangent of a sum or difference in simpler terms.

By substituting the values \( \alpha = \theta \) and \( \beta = \frac{\pi}{4} \), you can simplify complex expressions step by step using these identities. This strategic approach is crucial for breaking down challenging trigonometric problems into manageable parts.

Tangent Function

The tangent function, \( \tan(\theta) \), is one of the primary trigonometric functions. It represents the ratio of the sine to the cosine of the angle: \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \).

This function has unique properties, including a periodicity of \( \pi \), which means it repeats its values every \( \pi \) radians. It plays a pivotal role in various trigonometric identities, such as the tangent angle sum and difference identities.

When using formulas like \( \tan(\alpha + \beta) \) or \( \tan(\alpha - \beta) \), knowing the fundamental nature of tangent is helpful to understand how it interacts with different angles. For instance, in the problem posed, the tangent function is manipulated using these identities to transform and simplify expressions involving angle sums and differences.

Understanding the tangent function's behavior is key to solving trigonometric equations, especially when angles are involved. It always helps to visualize the tangent function on the unit circle to get a better grasp of its properties, including how it relates to other trigonometric functions.

Simplification Techniques

Simplifying trigonometric expressions often involves a combination of algebraic manipulation and trigonometric identities. A clear understanding of both is necessary to tackle complex equations.

For instance, to simplify the expression:\( \frac{1 + \tan \theta}{1 - \tan \theta} - \frac{1 - \tan \theta}{1 + \tan \theta} \),follow these steps:

• Combine the fractions by finding a common denominator, here \( (1 - \tan \theta)(1 + \tan \theta) \).
• Simplify the resulting expression, noting any identities that might apply, like \( 1 - \tan^2 \theta = \cos^2 \theta \) when \( \cos(2\theta) \) is involved.

Using these techniques, the expression simplifies to\( \frac{2 \tan \theta}{1 - \tan^2 \theta} \).

This expression directly relates to the double angle formula for tangent, \( \tan(2\alpha) = \frac{2 \tan \alpha}{1 - \tan^2 \alpha} \), showing the result is indeed \( 2 \tan 2\theta \). Such simplification techniques make solving complex trigonometric problems much smoother.