Problem 25 - Comprehensive Trigonometry for IIT JEE Main and Advanced Rejaul Makshud MeGraw Hill

Question

Solve for \(x\) : \(\sec ^{-1}\left(\frac{x}{a}\right)-\sec ^{-1}\left(\frac{x}{b}\right)=\sec ^{-1} b-\sec ^{-1} a .\)

Step-by-Step Solution

Answer

The solution for \(x\) in the given equation is: \(x = \frac{ab( a \cos(b) - b \cos(a) )}{ a^2 - b^2 }\)

Step 1: Rewrite the equation using the property of inverse secant

The inverse secant \(\sec^{-1}(x)\) identity can be rewritten as \(\frac{1}{\cos(x)}\). Applying this to the entire equation gives:\n\[\frac{1}{\cos(\frac{x}{a})}-\frac{1}{\cos(\frac{x}{b})}=\frac{1}{\cos( b )}-\frac{1}{\cos( a )}\]

Step 2: Rationalize each expression

Multiply each fraction by the Cos of their denominator to get:\n\[ab( a \cos(b) - b \cos(a) ) = x ( a^2 - b^2 )\]

Step 3: Isolate x

Finally, solve for \(x\) by dividing through by \( ( a^2 - b^2 )\) to isolate \(x\), giving the final solution as: \[x = \frac{ab( a \cos(b) - b \cos(a) )}{ a^2 - b^2 }\]

Key Concepts

Trigonometric Identities

Trigonometric identities play a crucial role in understanding and solving inverse trigonometric functions, such as the inverse secant equation given in the problem. An inverse secant function, denoted as \( \sec^{-1}(x) \), is essentially the angle whose secant is \( x \). To simplify the manipulation of these functions, we use trigonometric identities. These identities redefine complex trigonometric expressions into more manageable forms.

In the provided exercise, the identity used is \( \sec(x) = \frac{1}{\cos(x)} \). This expression allows us to work more effectively with inverse secants. By rewriting \( \sec^{-1}(x) \) in terms of cosines, you break down the functions into smaller parts. This makes them easier to work with, especially when simplifying or solving equations. Such strategic transformations are vital when approaching problems involving trigonometric functions. They often reveal simpler paths to the solution than one might initially expect.

Rationalization

Rationalization involves eliminating radicals or irrational numbers from the denominator of a fraction. It simplifies expressions and is an essential technique for solving complex trigonometric equations. In our problem, after rewriting the equation using trigonometric identities, each term of the equation becomes a fraction involving cosines.

Here, the process of rationalization is applied by multiplying each side of the equation by the cosine of the respective angles derived in the trigonometric identity application. The key purpose of this step is to clear any complex denominators that arise, facilitating further simplification. This practice not only simplifies the equation but also aligns like terms, allowing for clearer isolation of variables later in the solution.

In this particular step, by clearing the denominators, it allows us to express the equation without fractions, greatly simplifying the algebraic manipulation needed in subsequent steps.

Isolating Variables

Isolating the variable \( x \) is the final goal in solving the given trigonometric equation. This technique involves rearranging the equation so that \( x \) stands alone on one side of the equation. Once the expressions are rationalized, isolating \( x \) becomes more straightforward.

In practice, the solution requires dividing both sides of the equation by the expression \( (a^2 - b^2) \). This step enables you to single out \( x \), providing a clear solution to the problem. It's crucial to perform operations equally on both sides of the equation to maintain the balance between them.

Understanding how to effectively isolate variables is a fundamental skill in algebra and trigonometry. It is often the concluding step in solving equations, giving a concise and definitive answer to the problem at hand.