Problem 17 - Comprehensive Trigonometry for IIT JEE Main and Advanced Rejaul Makshud MeGraw Hill

Question

If \(x=\sec \theta-\tan \theta\) and \(y=\operatorname{cosec} \theta+\cot \theta\), then prove that \(x y+x-y+1=0\).

Step-by-Step Solution

Answer

After substituting the trigonometric identities into the given equations of \(x\) and \(y\), adding them up, simplifying and rearranging, \(x y + x - y + 1 = 0\). Thus, the given expression is proven.

Step 1: Substitute the Trigonometric Identities

Substitute the identity for \(\sec \theta, \tan \theta, \operatorname{cosec} \theta, \cot \theta \) into the equations of \(x\) and \(y\), yielding \(x = \frac{1}{\cos \theta} - \frac{\sin \theta}{\cos \theta}\) and \(y = \frac{1}{\sin \theta} + \frac{\cos \theta}{\sin \theta}\). Multiply these by \(\cos \theta\) and \(\sin \theta\) respectively, resulting in \(x \cos \theta = 1 - \sin \theta\) and \(y \sin \theta = 1 + \cos \theta\).

Step 2: Add the Equations

Add the two equations: \(x \cos \theta + y \sin \theta = 2 + \cos \theta - \sin \theta\). Based on the Pythagorean theorem, replace the \(\cos \theta\) with \(1 - \sin^2 \theta\) and the \(\sin \theta\) with \(1 - \cos^2 \theta\). By simplifying the equation further, it results in \(x \cos \theta + y \sin \theta = 2 - y - x\)

Step 3: Rearrange the Equation

Rearrange the last equation to form the requested equation. Simply add \(x + y - 1\) to the both sides of equation resulting in \(x \cos \theta + y \sin \theta + x + y - 1 = 1\), which simplifies to \(x y + x - y + 1 = 0\). This is the desired equation.

Key Concepts

Secant

The secant of an angle \(\theta\) in trigonometry, noted as \(\sec \theta\), represents the reciprocal of the cosine function. Put simply, it is \( \sec \theta = \frac{1}{\cos \theta} \). This definition means that the secant function becomes undefined wherever the cosine of \(\theta\) is zero. Understanding where these points occur is crucial, especially when solving equations or identities involving secant.

When visualizing the unit circle, the secant forms a segment from the origin to a point which intersects the circle extended from the \(x\)-axis. It's important to grasp how secant links to cosine; sector produces larger values as \(\theta\) approaches \(90^{\circ}\) or \(270^{\circ}\) due to the denominator nearing zero, creating asymptotes in its graph.

Tangent

The tangent of an angle \(\theta\), represented as \(\tan \theta\), is a function stemming from the sine and cosine ratios. It is computed as the ratio of sine to cosine, or \( \tan \theta = \frac{\sin \theta}{\cos \theta} \). This trigonometric function presents a periodic behavior with undefined values at odd multiples of \(90^{\circ}\), where \(\cos \theta = 0\).

Tangent functions often model angles of elevation and depression because of how they relate the opposite and adjacent sides of a right triangle. In the exercise provided, understanding the interplay between tangent and secant allows rearrangement and simplification of expressions and solutions. Bear in mind, when solving equations, to consider where it's undefined, as it hugely impacts the solutions.

Cosecant

Cosecant, noted as \(\operatorname{cosec} \theta\), is yet another critical element in trigonometry. It represents the reciprocal of the sine function. Therefore, \(\operatorname{cosec} \theta = \frac{1}{\sin \theta}\). As a result, this function is undefined where sine equals zero, specifically at integer multiples of \(180^{\circ}\).

In practical use, the cosecant function helps solve problems involving longer triangle sides, stretching inverse to the sine value but continuously needing alertness to points of discontinuation. These discontinuities are key in solving exercises that involve simplifying expressions to prove trigonometric identities, as they affect the outcome and value placement of such expressions.

Cotangent

The cotangent function, written as \(\cot \theta\), arises as the complement to tangent, being the reciprocal of tangent. Thus, \(\cot \theta = \frac{\cos \theta}{\sin \theta}\). It is undefined at points where sine is zero, coinciding with cotangent's role as tangent's reciprocal.

Exploring cotangent aids in understanding angles and side length ratios in right triangles from a unique perspective compared to tangent. Within the framework of exercises similar to the one provided, recognizing cotangent's behavior, especially around points of undefined values, is essential. This highlights the need to transform expressions skillfully to arrive at simplified equations or verified identities.