Problem 26 - Comprehensive Trigonometry for IIT JEE Main and Advanced Rejaul Makshud MeGraw Hill

Question

Solve the following equations and tick the correct one. If \(\tan 2 x \cdot \tan x=1\), then \(x\) is (a) \(\frac{\pi}{3}\) (b) \((6 n \pm 1) \frac{\pi}{6}\) (c) \((4 n \pm 1) \frac{\pi}{6}\) (d) \((2 n \pm 1) \frac{\pi}{6}\)

Step-by-Step Solution

Answer

The correct answer is (d).

Step 1: Simplify the given equation

We will start by simplifying the given equation. The formula to convert double angles in terms of single angles for tan can be used here. The formula is \(\tan 2a = \frac{2\tan a}{1 - \tan^2 a}\). Substitute \(\tan 2x\) using this formula. We get \(\frac{2 \tan x}{1 - \tan^2 x} \cdot \tan x=1\) or \(\frac{2 \tan^2 x}{1 - \tan^2 x}=1.\)

Step 2: Solve the equation further

Next step will be to solve this equation for \(\tan x\). After further simplification, we get \(\tan^2 x = \frac{1}{2}\) or \(\tan x = \pm\sqrt{1/2}\). This implies that \(x\) can be \(n\pi \pm \frac{\pi}{4}\).

Step 3: Match against the options

The last step is to match the values of \(x\) with the available options. Looking at the options, we find a match with the option (d) \((2 n \pm 1) \frac{\pi}{6}\), where \(n\) is an integer.

Step 4: Verify

To ensure the option (d) is correct, we can also put the value of \(x\) from (d) back into the equation to verify it. After substituting the values, it can be verified that both sides of equation matches. Hence, the solution for \(x\) is indeed (d).

Key Concepts

tan double angle formula

The tan double angle formula is a crucial concept in trigonometry. It allows us to express the tangent of a double angle in terms of the tangent of a single angle. This formula is given by:

• \( \tan 2a = \frac{2 \tan a}{1 - \tan^2 a} \)

This equation is particularly useful when solving trigonometric equations because it allows us to simplify expressions that involve double angles.
For instance, in the exercise given, we substitute \( \tan 2x \) using this formula. Doing so helps us convert the equation into a manageable form that is easier to solve. Understanding this formula can also help you recognize when an equation's complexity can be reduced.
Remember, the formula can only be applied when the expression matches its structure, so look for a double angle and a single angle in your equation.

solving equations

Solving equations involving trigonometric functions requires substituting known identities and simplifying the given expressions. In our exercise, after using the tan double angle formula, we obtained the equation:

• \( \frac{2 \tan^2 x}{1 - \tan^2 x} = 1 \)

To solve this equation:

• Isolate the trigonometric function \( \tan x \).
• Simplify the expression to find \( \tan^2 x \).
• Finally, solve for \( \tan x \).

The simplification led us to:

• \( \tan^2 x = \frac{1}{2} \)
• This simplifies to \( \tan x = \pm \sqrt{\frac{1}{2}} \).

Understanding how to manipulate and simplify trigonometric equations is essential for solving problems efficiently. It aids in identifying where and when to apply identities, rounding off the solution process.

trigonometric identities

Trigonometric identities are fundamental tools in the simplification and solution of trigonometric equations. They are equations involving trigonometric functions that are true for all values of the variable for which they are defined. Here are a few key identities frequently used:

• \( \tan^2 a + 1 = \sec^2 a \)
• \( \sin^2 a + \cos^2 a = 1 \)
• \( 1 + \cot^2 a = \csc^2 a \)

In the context of our exercise, knowing these identities allowed us to rewrite and simplify the given trigonometric equation. For instance, leveraging the knowledge that \( \tan 2x \) can be expressed using trigonometric identities helped reduce the complexity of the equation we were solving.
By recognizing which identities match the given components of a problem, you can transform unwieldy expressions into simpler ones that pave the way for finding solutions.