Problem 49 - 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 is verified as both sides simplify to 1.

Step 1: Transform the Left Side

We begin by recalling that \(1 - \cos^2 X = \sin^2 X\). Therefore, we can rewrite the left side of the equation as \( \sin^2 X (1 + \cot^2 X)\).

Step 2: Apply the Pythagorean Identity for Cotangent

Recall that \( \cot^2 X = \frac{\cos^2 X}{\sin^2 X} \). This means \( 1 + \cot^2 X = 1 + \frac{\cos^2 X}{\sin^2 X} = \frac{\sin^2 X + \cos^2 X}{\sin^2 X} \).

Step 3: Simplify the Expression

Recall the Pythagorean identity, which states \(\sin^2 X + \cos^2 X = 1\). Thus, \( \frac{1}{\sin^2 X} \) simplifies to \( \csc^2 X \).

Step 4: Evaluate the Expression

The expression \( \sin^2 X \cdot \csc^2 X \) simplifies to \( 1 \), because \( \csc^2 X = \frac{1}{\sin^2 X} \) and \( \sin^2 X \times \frac{1}{\sin^2 X} = 1 \).

Step 5: Identity Verification

By simplifying both sides, the left side of the given identity equals the right side, \( 1 \). Therefore, the original equation is a valid identity.

Key Concepts

Pythagorean identity

The Pythagorean identity is a fundamental concept in trigonometry that simplifies the study of trigonometric functions. At its core, it stems from the Pythagorean theorem, which in the context of trigonometry, translates to the identity: \( \sin^2 X + \cos^2 X = 1 \). This expression highlights the intrinsic relationship between the sine and cosine functions. If you know the value of either \( \sin X \) or \( \cos X \), you can easily determine the other using this identity.

This identity is particularly useful when transforming and simplifying trigonometric expressions. For example, if you want to find \( 1 - \cos^2 X \), you can use the Pythagorean identity to rewrite it as \( \sin^2 X \). This kind of transformation simplifies complex expressions and helps solve trigonometric equations.

• The identity emphasizes a core geometric aspect of the unit circle, where the radius is always 1.
• It is applicable in various branches of mathematics, not just trigonometry, including calculus and physics.
• Remember that the identity holds true for any angle \( X \), making it incredibly versatile.

cotangent function

The cotangent function, denoted as \( \cot X \), is one of the six primary trigonometric functions. It is the reciprocal of the tangent function, and is defined as \( \cot X = \frac{1}{\tan X} = \frac{\cos X}{\sin X} \). Understanding the cotangent function is pivotal in the context of trigonometric identities due to its relational properties.

In the above exercise, using the expression \( \cot^2 X = \frac{\cos^2 X}{\sin^2 X} \) allowed us to manipulate the equation into a useful form. Rewriting \( 1 + \cot^2 X \) simplifies to \( \frac{\sin^2 X + \cos^2 X}{\sin^2 X} \), showcasing the recurrent presence of the Pythagorean identity.

• The cotangent function is undefined when \( \sin X = 0 \), as division by zero is impossible.
• It is periodic with a period of \( \pi \), similar to tangent but offset by phase shifts.
• Cotangent values are positive in the first and third quadrants, where sine and cosine share signs.

cosecant function

The cosecant function, represented as \( \csc X \), is another key reciprocal trigonometric function. It is the reciprocal of the sine function defined as \( \csc X = \frac{1}{\sin X} \). In trigonometric identities and transformations, it often appears when expressions involve division by \( \sin X\).

In solving the provided identity, recognizing \( \csc^2 X \), which is \( \frac{1}{\sin^2 X} \), allowed for further simplification of the expression \( \sin^2 X \cdot \csc^2 X \) to 1. This demonstrates how reciprocal functions can elegantly simplify seemingly complex trigonometric expressions.

• The cosecant is undefined at points where \( \sin X = 0 \), similar to how division by zero is undefined.
• \( \csc X \) has the same period as the sine function, \( 2\pi \).
• It is useful in solving problems involving right-angled triangles and when working with ratios of a circle.