Problem 48 - Comprehensive Trigonometry for IIT JEE Main and Advanced Rejaul Makshud MeGraw Hill

Question

If \(p>q>0, p r<-1

Step-by-Step Solution

Answer

Yes, the equation is proved to be true.

Step 1: Start with the addition of first two inverse tangents

Using the addition formula for two inverse tangents with \(x=\frac{p-q}{1+pq}\) and \(y=\frac{q-r}{1+qr}\), we get: \(\tan^{-1}\left(\frac{p-q}{1+pq}\right) + \tan^{-1}\left(\frac{q-r}{1+qr}\right) = \tan^{-1} \left(\frac{\frac{p-q}{1+pq} + \frac{q-r}{1+qr}}{1 - \frac{p-q}{1+pq}\frac{q-r}{1+qr}}\right)\).

Step 2: Simplify the expression

By simplifying the expression inside the inverse tangent function, we get: \(\tan^{-1}\left(\frac{p-r}{1-pr}\right)\). Therefore, \(\tan^{-1}\left(\frac{p-q}{1+pq}\right)+\tan^{-1}\left(\frac{q-r}{1+qr}\right)=\tan^{-1}\left(\frac{p-r}{1-pr}\right).

Step 3: Add the last inverse tangent

We return to the original equation and add the last term on both sides: \(\tan^{-1}\left(\frac{p-q}{1+pq}\right)+\tan^{-1}\left(\frac{q-r}{1+qr}\right)+\tan^{-1}\left(\frac{r-p}{1+rp}\right) = \tan^{-1}\left(\frac{p-r}{1-pr}\right) + \tan^{-1}\left(\frac{r-p}{1+rp}\right).

Step 4: Use the addition formula again

Apply the addition formula again to the right-hand side of this equation, with \(x=p-r\) and \(y=r-p\). We can simplify the expression to result in \(\tan^{-1}(\infty)\). As the tangent function approaches \(\pi/2\) when its argument approaches \(\infty\), this results in the proof that \(\tan^{-1}\left(\frac{p-q}{1+pq}\right)+\tan^{-1}\left(\frac{q-r}{1+qr}\right)+\tan^{-1}\left(\frac{r-p}{1+rp}\right)=\pi.\)

Key Concepts

Addition Formula for Inverse Tangent

The addition formula for inverse tangents is an essential concept to understand when working with trigonometric equations, particularly when dealing with the tangent inverse or arctan function.
When you have two angles represented as inverse tangents, here's how you combine them:

• If \(x = \tan^{-1}(a)\) and \(y = \tan^{-1}(b)\), then the sum \(\tan^{-1}(a) + \tan^{-1}(b) = \tan^{-1}\left(\frac{a + b}{1 - ab}\right)\) given that \(ab < 1\).
• This formula is pivotal in simplifying expressions involving the sum of inverse tangents.

In our original exercise, applying this formula allows us to simplify the sum of two inverse tangents to a single inverse tangent. This step narrows down the complexity of the problem and opens paths for further simplifications.
Using these formulas correctly can transform challenging expressions into more manageable ones, aligning with the properties of tangent and its inverse.

Trigonometric Identities

Trigonometric identities form the backbone of resolving equations involving trigonometric functions. These identities are equalities involving trigonometric functions that are true for all values within their respective domains.
In particular, the tangent identities are crucial here:

• \(\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}\), which relates the tangent function to sine and cosine.
• Inverse tangent identities include \(\tan^{-1}(-a) = -\tan^{-1}(a)\).
• The sum and difference of angles identity lets us explore different combinations: \(\tan(a \pm b) = \frac{\tan a \pm \tan b}{1 \mp \tan a \tan b}\).

Identifying and applying these identities correctly is crucial in collapsing complex expressions into simpler forms, paving the way to solve problems that initially seem intricate or insurmountable.
In the context of our task, these identities help in translating and transforming the given expressions of inverse tangents into forms that are more accessible for further algebraic manipulations.

Inequalities in Trigonometry

Inequalities are mathematical expressions that show how quantities relate to one another. With trigonometry, these inequalities often involve trigonometric functions in various forms. They offer a way to compare magnitudes or determine limits for variables.
Consider our context in the given problem:

• The conditions \(p > q > 0\) and \(p \cdot r < -1 < q \cdot r\) create a framework under which the functions are evaluated. These inequalities guide what assumptions can be inferred about the values of the trigonometric functions and their inverses.

By working within these inequalities, one can explore the feasibility of certain transformations or simplifications.
This is essential for ensuring that the properties and results used are valid and that their use will lead to a meaningful result within the constraint system set by the problem statement.