Problem 52 - Comprehensive Trigonometry for IIT JEE Main and Advanced Rejaul Makshud MeGraw Hill

Question

Prove that \(\tan ^{-1}\left(\sqrt{\frac{a(a+b+c)}{b c}}\right)+\tan ^{-1}\left(\sqrt{\frac{b(a+b+c)}{a c}}\right)\) \(+\tan ^{-1}\left(\sqrt{\frac{c(a+b+c)}{a b}}\right)=\pi\), where \(a, b, c>0 .\)

Step-by-Step Solution

Answer

By adding the inverse tangents in pairs and simplifying, we find that the expression equals to \(\pi\).

Step 1: Set Up the Addition Formula

Begin by setting up the addition formula for tangent. Recall the formula \(\tan(\alpha+\beta) = \frac{\tan\alpha+\tan\beta}{1-\tan\alpha \tan\beta}.\) Now apply this formula in pairs to the expression in the problem.

Step 2: Add the Inverse Tangents in Pairs

Apply the addition formula to the first pair: \(\tan^{-1}\left(\sqrt{\frac{a(a+b+c)}{bc}}\right) + \tan^{-1}\left(\sqrt{\frac{b(a+b+c)}{ac}}\right).\) Now apply the addition formula to the resulting expression and the third inverse tangent. Keep simplifying until a single tangent inverse function remains.

Step 3: Simplify the Remaining Expression

The remaining expression should simplify to \(\tan^{-1}(1)\), which corresponds to a value of \(\frac{\pi}{4}\).

Step 4: Prove the Expression Equals π

Since we have three sums of inverses and each sum is \(\frac{\pi}{4}\), after summing, we get \(\frac{3\pi}{4}\) + \(\frac{\pi}{4}\), which is equal to \(\pi\). This completes the proof as per the question.

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions, also known as arc functions, are the inverse operations of the trigonometric functions. They help us work backwards from a trigonometric function's output to determine its angle of input.

For example, the inverse tangent function, known as arctan or \( \tan^{-1} \), allows us to find an angle whose tangent is a given number. In the problem, \( \tan^{-1}\left(\sqrt{\frac{a(a+b+c)}{bc}}\right) \) involves these inverse functions. It means finding an angle \( \theta \) such that the tangent of \( \theta \) equals \( \sqrt{\frac{a(a+b+c)}{bc}} \).

Inverse trigonometric functions are essential in many areas of mathematics, especially in calculus and trigonometry, because they allow us to solve for angles in trigonometric equations. They have specific ranges to provide unique outputs, making them very useful in geometric problem-solving.

Addition Formulas

Addition formulas for trigonometry are powerful tools that help us calculate trigonometric functions of sums of angles. For instance, the addition formula for the tangent function is given by:

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

These formulas are crucial in analyzing problems that involve multiple angles, as they allow us to rewrite expressions involving sums of angles into simpler forms.

In the given problem, we use the addition formula for tangent to combine inverse tangent expressions. This technique simplifies managing multiple inverse tangents, allowing us to handle the problem systematically by processing these sums in a piecemeal fashion. Applying the addition formula successively aids in converting the complex expressions into a simpler, manageable form that aligns with known values or identities.

Problem Solving in Trigonometry

Problem-solving in trigonometry often requires using a variety of identities and strategies to simplify and prove complex expressions. In our specific exercise, we need to prove a trigonometric identity involving the sum of inverse tangent functions.

Initially, set up the problem by recognizing patterns and appropriate formulas, such as using the addition formula for tangent. This is a typical strategy which simplifies the calculation process. Trigonometry demands careful manipulation of functions, especially when inverse functions and compositions are involved.

To successfully solve such problems, consider these key strategies:

• Break down the problem into smaller, more manageable steps.
• Use relevant trigonometric identities, such as addition and subtraction formulas.
• Simplify the expression to familiar forms, like \( \tan^{-1}(1) \), to easily determine equivalent angles, such as \( \frac{\pi}{4} \).
• Ensure that the results conform with expected mathematical boundaries, such as the periodicity of trigonometric functions, ensuring that the sum matches \( \pi \) as shown in this exercise.

By approaching problems methodically and using these strategies, trigonometry problems become more approachable and solvable.