Problem 21 - Comprehensive Trigonometry for IIT JEE Main and Advanced Rejaul Makshud MeGraw Hill

Question

If \(u=\cot ^{-1}(\sqrt{\tan \alpha})-\tan ^{-1}(\sqrt{\tan \alpha})\), then \(\tan \left(\frac{\pi}{4}-\frac{u}{2}\right)\) is equal to (a) \(\sqrt{\tan \alpha}\) (b) \(\sqrt{\cot \alpha}\) (c) \(\tan \alpha\) (d) \(\cot \alpha\)

Step-by-Step Solution

Answer

Hence, the answer is (a) \(\sqrt{\tan \alpha}\).

Step 1: Simplify the expression

Let's denote \(X=\sqrt{\tan \alpha}\) to simplify the expression. Then \(u=\cot ^{-1}(X)-\tan ^{-1}(X)\). The first step would be to express the cotangent as the reciprocal of the tangent, so rewrite the expression as \(u=\tan ^{-1}(1/X)-\tan ^{-1}(X)\).

Step 2: Apply the tan-1(a) – tan-1(b) = tan-1((a-b)/(1+ab)) formula

Applying this formula, we get \(u=\tan^{-1}\left(\frac{1/X-X}{1+1}\right)\), which simplifies as \(u=\tan^{-1}(X-1/X)=\tan^{-1}\left(\frac{X^2-1}{X}\right)\).

Step 3: Apply the tan(π/4 - u/2) = tan(tan-1((1 - a^2)/(1 + a^2))) formula

We can then rewrite \(u\) to fit the tan of difference formula, and we get \(\tan \left(\frac{\pi}{4}-\frac{u}{2}\right)=\tan \left(\tan^{-1}\left(\frac{1-(X^2-1/X)^2}{1+(X^2-1/X)^2}\right)\right)\). Expand and simplify the expression inside the outer tangent to get \(\tan \left(\frac{\pi}{4}-\frac{u}{2}\right)=X\).

Step 4: Substitute X back to get the final answer

Substituting \(X=\sqrt{\tan \alpha}\) back in, we get \(\tan \left(\frac{\pi}{4}-\frac{u}{2}\right)=\sqrt{\tan \alpha}\).

Key Concepts

Cotangent Identities

In trigonometry, understanding the different trigonometric identities is crucial for simplifying and solving various expressions and equations. Among these identities, the cotangent identity is quite useful. The cotangent function is defined as the reciprocal of the tangent function, that is, \[ \cot \theta = \frac{1}{\tan \theta} \].
This simple relationship allows us to interconvert expressions involving cotangent and tangent, which can particularly be helpful when dealing with inverse trigonometric functions.

• The inverse cotangent function, \( \cot^{-1}(x) \), returns the angle whose cotangent is \(x\).
• By the reciprocal property, you can convert it using tangent: \( \cot^{-1}(x) = \tan^{-1}\left(\frac{1}{x}\right) \).

This identity plays a significant role in our problem where we express \[ u = \cot^{-1}(X) - \tan^{-1}(X) \] in terms of tangent identities, making it easier to simplify and solve.

Tangent Subtraction Formula

The tangent subtraction formula is another key tool in trigonometry that helps break down complex expressions involving angles. The formula is given by: \[ \tan(a - b) = \frac{\tan a - \tan b}{1 + \tan a \cdot \tan b} \].
This formula is critical in our exercise for simplifying expressions involving differences between inverse trigonometric functions. It specifically allows combining and reducing terms like \( \tan^{-1}(x) \) and \( \tan^{-1}(y) \) into a single \( \tan^{-1} \) term.
In the problem, we apply it directly to: \[ u = \tan^{-1}\left(\frac{1/X - X}{1 + 1}\right) \]which simplifies further into uniform tangent terms. Employing this approach is a strategic part of simplifying trigonometric expressions, resulting in an expression that can be evaluated more readily.

Trigonometric Simplification

Simplifying trigonometric expressions is often necessary to solve equations or find solutions to problems like the one in this exercise. Here are some steps and tips to master trigonometric simplification:

• Use known identities: Start by substituting known identities like cotangent and tangent relations to save steps and reduce complexity.
• Simplify inverse relations: Turning inverse functions using identities and subtraction formulas helps reduce the expression to a simpler form.
• Change variables if needed: Set variables (like \( X = \sqrt{\tan \alpha} \) in our problem) to simplify calculation and enable easier substitution back once simplified.
• Double-check expansions: Ensure that polynomial expressions derived from trigonometric identities are accurately expanded to avoid missteps.
• Relate back to original variable: Substitute back any auxiliary variable used to get an answer with respect to the initial problem variable.

These methods enable a clean path from a complex expression to a simpler form, allowing us to find that: \[ \tan \left(\frac{\pi}{4} - \frac{u}{2}\right) = X \] and consequently \( X = \sqrt{\tan \alpha} \), as simplified through careful calculation.