Problem 4 - Comprehensive Trigonometry for IIT JEE Main and Advanced Rejaul Makshud MeGraw Hill

Question

Solve the following equations and tick the correct one. If \(\tan ^{2} \theta+\cot ^{2} \theta=2\), then \(\theta\) is (a) \(n \pi+\frac{\pi}{6}\) (b) \(n \pi-\frac{\pi}{6}\) (c) \(n \pi+\frac{\pi}{4}\) (d) \(n \pi-\frac{\pi}{4}\)

Step-by-Step Solution

Answer

\(\theta = n\pi\), where \(n\) is any integer (none of the given options).

Step 1: Analyze the given identities

We know that the square of cotangent can be expressed in terms of tangent as \( \cot ^{2} \theta = \frac{1}{\tan^{2}\theta} \). And from the Pythagorean identity \( \tan^{2}\theta + 1 = \sec^{2}\theta \), we can express tangent in terms of secant as \( \tan^{2}\theta = \sec^{2}\theta - 1 \).

Step 2: Substitute the identities in the equation

Substitute these identities into the given equation. \( ( \sec^{2}\theta - 1) + \frac{1}{\sec^{2}\theta - 1} = 2 \).

Step 3: Simplify the equation

Upon simplifying the equation, we get \( \sec^{4}\theta - 2 = 2\sec^{2}\theta \).

Step 4: Rewrite the equation

Rewrite the equation as a quadratic equation \( \sec^{4}\theta - 2\sec^{2}\theta - 2 = 0 \).

Step 5: Solve for \(\sec^{2}\theta\)

Now, solve this quadratic equation in \( \sec^{2}\theta \). The solutions are \( \sec^{2}\theta = 1, -2 \). But \( \sec^{2}\theta \) cannot be negative. So, \( \sec^{2}\theta = 1 \).

Step 6: Solve for \(\theta\)

As we know that \( \sec\theta = \pm 1 \) which implies \(\cos\theta = \pm 1 \). Thus, \(\theta = n\pi \) where \(n\) is any integer.

Step 7: Check the options

Option (a) \(n \pi+\frac{\pi}{6}\), (b) \(n \pi-\frac{\pi}{6}\), (c) \(n \pi+\frac{\pi}{4}\), and (d) \(n \pi-\frac{\pi}{4}\), give a value of \(\cos\theta = \pm 1\) when \(\theta = n\pi\). Hence, \(\theta = n\pi\) is the correct solution.

Key Concepts

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for every value of the variable where both sides of the equation are defined. These identities are crucial tools in solving trigonometric equations. They allow us to simplify complex expressions and find solutions.

• Common Trigonometric Identities: There are several foundational identities like sine, cosine, tangent, secant, cosecant, and cotangent.
• Useful for Simplification: Identities such as the reciprocal identity (\( an heta = \frac{1}{ ext{cot} heta} \)) or the quotient identity (\( an heta = \frac{ ext{sin} heta}{ ext{cos} heta} \)) can transform trigonometric expressions.

In our original problem, we used the identity \( \tan^2 \theta + \cot^2 \theta = 2 \). By applying related identities, we can transform this equation into a more familiar form that can be solved in this manner.

Tangent and Cotangent

Tangent and cotangent are specific trigonometric functions, each with their own unique properties and identities. They are often used together since they are reciprocals of each other.

• Tangent Function: The tangent of an angle is defined as the ratio of the opposite side to the adjacent side in a right-angled triangle. In terms of sine and cosine, \( \tan \theta = \frac{\sin \theta}{\cos \theta} \).
• Cotangent Function: This is the reciprocal of the tangent function, \( \cot \theta = \frac{1}{\tan \theta} = \frac{\cos \theta}{\sin \theta} \).

In problems, knowing the relationship \( \tan^2 \theta \) and \( \cot^2 \theta \) can be very helpful as it allows conversion of equations into a solvable form such as transforming them using Pythagorean and reciprocal identities for simplification.

Pythagorean Identity

The Pythagorean identity is one of the most fundamental identities in trigonometry. It establishes a relationship between sine, cosine, and sometimes tangent.

• Basic Identity: The most basic form of the Pythagorean identity is \( \sin^2 \theta + \cos^2 \theta = 1 \).
• Related Identities: When using tangent, another form is \( \tan^2 \theta + 1 = \sec^2 \theta \). These are derived by dividing the basic identity by \( \cos^2 \theta \).

These identities are pivotal when solving equations that involve squared terms of sine, cosine, or tangent. In the exercise, the Pythagorean identity allows us to convert the given equation into a form that’s more straightforward to solve, specifically, by expressing \( \tan^2 \theta \) in terms of \( \sec^2 \theta - 1 \). This lays a foundation for using other identities to simplify and find the solution.