Problem 56 - Comprehensive Trigonometry for IIT JEE Main and Advanced Rejaul Makshud MeGraw Hill

Question

Solve the following equations and tick the correct one. If \(\tan (\pi \cos \theta)=\cot (\pi \sin \theta)\), then the value of \(\cos \left(\theta-\frac{\pi}{4}\right)\) is (a) \(\frac{1}{2}\) (b) \(\frac{1}{\sqrt{2}}\) (c) \(\frac{1}{2 \sqrt{2}}\) (d) None

Step-by-Step Solution

Answer

The answer is \((d)\) None. The value of \(\cos \left(\theta-\frac{\pi}{4}\right)\) is 1, which is not an option in the given answers.

Step 1: Use cotangent identity

Start by replacing the \(\cot\) function by its equivalent \(\frac{1}{\tan}\). This gives us the equation:\[ \tan(\pi \cos \theta) = \frac{1}{\tan(\pi \sin \theta)}\]

Step 2: Perform Cross-Multiplication

Perform a cross-multiplication and get, \[ \tan(\pi \cos \theta) \cdot \tan(\pi \sin \theta) =1 \] The property of tan and cotan makes this equation true only when \( \cos\theta = \sin \theta \), and this happens when \( \theta = \frac{\pi}{4} \).

Step 3: Replace theta in the target expression

After determining that \( \theta = \frac{\pi}{4} \), replace \(\theta\) by this value in the given expression \(\cos \left(\theta-\frac{\pi}{4}\right)\).This gives:\[ \cos \left( \frac{\pi}{4}-\frac{\pi}{4} \right) = \cos(0) \]

Step 4: Calculate Cosine Value

The cosine of zero is 1, thus\[ \cos(0) = 1 \]

Key Concepts

Cotangent identity

Understanding the cotangent identity is crucial when solving trigonometric equations. The cotangent function (denoted as \( \cot \)) is defined as the reciprocal of the tangent function. Simply put:

• \( \cot(x) = \frac{1}{\tan(x)} \)

This identity allows us to transform equations involving cotangent into those involving tangent, which can often simplify the problem. By rewriting \( \cot(\pi \sin \theta) \) as \( \frac{1}{\tan(\pi \sin \theta)} \), we can then more easily compare the expressions. Recognizing and using identities like this helps in breaking down and solving complex trigonometric equations effectively.

Value of cosine function

The cosine function, represented as \( \cos \), is one of the fundamental trigonometric functions. It describes the ratio of the adjacent side to the hypotenuse in a right-angle triangle. Evaluating this function for specific angles can help deduce solutions in trigonometric equations.
In the context of this problem, once \( \theta = \frac{\pi}{4} \) is determined, we need to find \( \cos \left(\theta - \frac{\pi}{4}\right) \). Substituting the known value gives us:

• \( \cos \left(\frac{\pi}{4} - \frac{\pi}{4}\right) = \cos(0) \)
• The cosine of zero is consistently known as 1, consequently leading to \( \cos(0) = 1 \).

Understanding how to compute these for common angles is essential in trigonometry, as these values form the basis of many advanced calculations.

Cross-multiplication in trigonometry

Cross-multiplication is a powerful algebraic tool, also applicable to trigonometry. It involves multiplying across the equality in an equation to simplify and potentially solve it. Here’s how it was used in this problem:

• We start with an equation \( \tan(\pi \cos \theta) = \frac{1}{\tan(\pi \sin \theta)} \).
• Cross-multiplying yields \( \tan(\pi \cos \theta) \cdot \tan(\pi \sin \theta) = 1 \).

This step is vital because it offers us a neat equation to analyze. Often, such relations point to an underlying principle or identity that can be exploited.
In this example, the product of these tangents equaling one indicates that \( \cos \theta \) must equal \( \sin \theta \), which is true at \( \theta = \frac{\pi}{4} \). Thus, cross-multiplication not only aids in simplifying but also in revealing critical insights about the equation.