Problem 47 - Algebra and Trigonometry

Question

Verify the identity. $$ \left(1-\cos ^{2} x\right)\left(1+\cot ^{2} x\right)=1 $$

Step-by-Step Solution

Answer

The identity \((1-\cos^2 x)(1+\cot^2 x) = 1\) is verified as true.

Step 1: Expand the given expression

The given identity is \((1 - \cos^2 x)(1 + \cot^2 x) = 1\). We will first expand the left side. Notice that \(1 - \cos^2 x\) is equal to \(\sin^2 x\) because of the Pythagorean identity \(\sin^2 x + \cos^2 x = 1\). So, substitute \(\sin^2 x\) for \((1 - \cos^2 x)\). The expression becomes \(\sin^2 x (1 + \cot^2 x)\).

Step 2: Substitute for \(\cot^2 x\)

We know the identity \(\cot^2 x = \frac{\cos^2 x}{\sin^2 x}\). Substitute this into the expression from Step 1. The expression becomes: \[ sin^2 x \left(1 + \frac{\cos^2 x}{\sin^2 x}\right) \] Simplify inside the parentheses to get:\[ \sin^2 x \left(\frac{\sin^2 x + \cos^2 x}{\sin^2 x}\right) \]

Step 3: Simplify the fraction

Since \(\sin^2 x + \cos^2 x = 1\), this simplifies the expression within the parenthesis to \(\frac{1}{\sin^2 x}\). The expression now becomes:\[ \sin^2 x \times \frac{1}{\sin^2 x} \] which simplifies to 1.

Step 4: Verify the identity

Now that the left side simplifies to 1, it matches the right side of the equation. This verifies that the given identity \((1 - \cos^2 x)(1 + \cot^2 x) = 1\) is true.

Key Concepts

Pythagorean identity

The Pythagorean identity is a fundamental concept in trigonometry that arises from the Pythagorean theorem. The identity \[ \sin^2 x + \cos^2 x = 1 \] is a powerful tool that expresses the relationship between the sine and cosine of an angle in a right triangle.
It essentially states that for any angle \(x\), the square of its sine plus the square of its cosine will always equal one. This identity forms the basis for many trigonometric derivations and simplifications.

• One important application is transforming expressions, as seen in the exercise when \(1 - \cos^2 x\) was replaced by \(\sin^2 x\).
• Students often use this identity to solve equations and verify other identities, making it a cornerstone of trigonometric problem-solving.

Recognizing and applying the Pythagorean identity correctly can simplify complex trigonometric expressions and is essential for verifying identities like the one in your exercise.

cotangent

Cotangent, commonly denoted as \( \cot x \), is a trigonometric function that is the reciprocal of the tangent function. It is defined as the ratio of the cosine to the sine of an angle. The relationship is given by the expression:\[ \cot x = \frac{\cos x}{\sin x} \]This function is particularly useful in simplifying expressions and solving trigonometric equations.
In the exercise, you needed to simplify \(1 + \cot^2 x\). Using the identity \( \cot^2 x = \frac{\cos^2 x}{\sin^2 x} \), the task becomes manageable.
This substitution allows the expression to be rewritten in a more recognizable form that can be further simplified using the Pythagorean identity.
Understanding cotangent and its properties is crucial when working with trigonometric identities. It allows us to transform and manipulate expressions, especially in verification exercises like yours. Remembering that \( \cot x = 1/\tan x \) can also be helpful when switching between related trigonometric functions.

simplification of expressions

Simplifying expressions involves transforming a complex expression into a simpler or more easily manageable form. In trigonometry, simplification often requires using identities and algebraic manipulations.
The primary goal is to rewrite expressions in a way that makes them easier to understand or solve.
In your exercise, simplification was key to verifying the identity.

• First, the expression \((1 - \cos^2 x)\) was simplified using the Pythagorean identity to become \(\sin^2 x\).
• Next, substituting \(\cot^2 x\) with \(\frac{\cos^2 x}{\sin^2 x}\) allowed further simplification by recognizing fractions that equated to 1.

The process of simplification can often reveal insights or hidden patterns within mathematical expressions. For students, practicing simplification helps build strong analytical skills and enhances understanding of how different mathematical concepts interrelate.
Knowing various identities, like the Pythagorean identity and trigonometric definitions, is essential in this process.