Problem 30 - 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 values of \(\cos \left(\theta-\frac{\pi}{4}\right)\) is (are) (a) \(\frac{1}{2}\) (b) \(\frac{1}{\sqrt{2}}\) (c) \(\pm \frac{1}{2 \sqrt{2}}\) (d) None

Step-by-Step Solution

Answer

(d) None

Step 1: Convert cot to tan

Cotangent is the reciprocal of tangent, rewrite \(\cot (\pi \sin \theta)\) as \(\frac{1}{\tan (\pi \sin \theta)}\). The given equation becomes \(\tan (\pi \cos \theta) = \frac{1}{\tan (\pi \sin \theta)}\).

Step 2: Simplify the equation

Cross multiply to get \(\tan (\pi \cos \theta) \tan (\pi \sin \theta) = 1\). As we know that \(\tan x \tan y = 1\) if \(x + y = \frac{\pi}{4}\), so we have \(\pi \cos \theta + \pi \sin \theta = \frac{\pi}{4}\), which simplifies to \(\cos \theta + \sin \theta = \frac{1}{4}\).

Step 3: Find the value of theta

Solving the equation \(\cos \theta + \sin \theta = \frac{1}{4}\), we get the solution as \(\theta = \frac{\pi}{4}\).

Step 4: Substitute the value of theta in the second equation

Now, we substitute \(\theta = \frac{\pi}{4}\) in the second equation \(\cos \left(\theta-\frac{\pi}{4}\right)\), so \(\cos \left( \frac{\pi}{4} - \frac{\pi}{4}\right)\) = \(\cos (0)\) = 1, but none of the given options match this value.

Key Concepts

Solving Trigonometric Equations

Solving trigonometric equations involves finding unknown values that make the equation true. In trigonometry, this might mean expressing angles or using trigonometric identities to simplify the equation. In the given original exercise, the equation was

• \( \tan (\pi \cos \theta) = \cot (\pi \sin \theta) \).

To solve this, the equation was first transformed using the identity \( \cot x = \frac{1}{\tan x} \), leading to

• \( \tan (\pi \cos \theta) = \frac{1}{\tan (\pi \sin \theta)} \).

Once the equation is rewritten, the next step is to cross-multiply to obtain \( \tan (\pi \cos \theta) \tan (\pi \sin \theta) = 1 \). Solving such equations often involves rearranging terms, using algebraic identities, and making logical deductions based on known trigonometric properties.

Trigonometric Functions

Trigonometric functions are the backbone of many mathematical problems involving triangles and periodic phenomena. The six basic trigonometric functions include sine, cosine, tangent, cosecant, secant, and cotangent. Each function relates an angle in a right-angled triangle to the ratio of two of its sides.

Cotangent \( \cot(x) \) and tangent \( \tan(x) \) are reciprocal functions, meaning that cotangent is the inverse of tangent. This relationship is crucial in simplifying trigonometric equations, as seen in the given problem. Understanding these reciprocal relationships helps in rewriting and solving trigonometric identities and equations effectively. These basic mathematical tools are essential for expanding and simplifying expressions, allowing for easier manipulation and solution of equations.

Angle Subtraction Formula

The angle subtraction formula is a key trigonometric identity that allows you to express the cosine of the difference of two angles in terms of their individual sines and cosines. It is expressed as:

• \( \cos(a - b) = \cos a \cos b + \sin a \sin b \).

In the exercise provided, the angle subtraction formula helps find the cosine of \( \theta - \frac{\pi}{4} \). By substituting \( \theta = \frac{\pi}{4} \), the expression becomes \( \cos \left( 0 \right) \), which simplifies to 1.

However, given options didn't match this result, indicating that careful verification of solution steps and cross-checking initial assumptions can be crucial. The angle subtraction formula serves as a powerful tool in trigonometry, particularly in solving equations where angles need to be expressed in terms of simpler components.