Problem 35 - Precalculus

Question

Prove or disprove the identity. $$\csc ^{2} x\left(1+\sin ^{2} x\right)=\cot ^{2} x$$

Step-by-Step Solution

Answer

The identity is disproven.

Step 1: Express Identity in Terms of Basic Functions

The original identity is given as \( \csc^2 x (1 + \sin^2 x) = \cot^2 x \). To begin, express \( \csc^2 x \) and \( \cot^2 x \) in terms of \( \sin x \) and \( \cos x \). Recall that \( \csc x = \frac{1}{\sin x} \) and \( \cot x = \frac{\cos x}{\sin x} \). Thus, \( \csc^2 x = \frac{1}{\sin^2 x} \) and \( \cot^2 x = \frac{\cos^2 x}{\sin^2 x} \).

Step 2: Substitute and Simplify Left Side

Substituting these into the left side of the identity, we have \( \csc^2 x (1 + \sin^2 x) = \frac{1}{\sin^2 x} (1 + \sin^2 x) \). This simplifies to \( \frac{1 + \sin^2 x}{\sin^2 x} \) or \( \frac{1}{\sin^2 x} + 1 \).

Step 3: Further Simplify Left Side

The expression \( \frac{1}{\sin^2 x} + 1 \) can be expanded to \( \frac{1 + \sin^2 x}{\sin^2 x} = \frac{1}{\sin^2 x} + \frac{\sin^2 x}{\sin^2 x} = \frac{1}{\sin^2 x} + 1 \). This simplifies directly to \( \csc^2 x \) because \( \frac{1}{\sin^2 x} = \csc^2 x \), so this part simplifies to \( \csc^2 x \) itself.

Step 4: Assess the Right Side

Now, evaluate the right side, \( \cot^2 x = \frac{\cos^2 x}{\sin^2 x} \).

Step 5: Compare Both Sides of the Identity

The left side simplifies to \( \frac{1}{\sin^2 x} + 1 \), while the right side is \( \frac{\cos^2 x}{\sin^2 x} \). Recognizing the Pythagorean identity \( \sin^2 x + \cos^2 x = 1 \), rewrite \( \frac{1}{\sin^2 x} + 1 \) by expressing 1 as \( \frac{\sin^2 x + \cos^2 x}{\sin^2 x} \). This makes the left side \( \frac{1 + \cos^2 x}{\sin^2 x} \), which does not simplify to \( \frac{\cos^2 x}{\sin^2 x} \). Therefore, the original identity, \( \csc^2 x (1 + \sin^2 x) = \cot^2 x \), does not hold true.

Key Concepts

Cosecant Function

The cosecant function, often denoted as \( \csc x \), is one of the basic trigonometric functions related to the sine function. It is defined as the reciprocal of the sine function. This means:

• \( \csc x = \frac{1}{\sin x} \)

The cosecant of an angle in a right triangle is the ratio of the length of the hypotenuse to the length of the side opposite the angle. Because it is based on the sine, it is undefined for angles where the sine is zero, like multiples of \( \pi \) (e.g., 0, \( \pi \), 2\( \pi \), etc.). When dealing with squares of trigonometric functions, like \( \csc^2 x \), it becomes:

• \( \csc^2 x = \left( \frac{1}{\sin x} \right)^2 = \frac{1}{\sin^2 x} \)

The cosecant function is essential in various trigonometric identities and plays a crucial role in analyzing periodic phenomena.

Cotangent Function

The cotangent function, represented by \( \cot x \), is another fundamental trigonometric function. It is the reciprocal of the tangent function, thus:

• \( \cot x = \frac{1}{\tan x} = \frac{\cos x}{\sin x} \)

The cotangent of an angle is the ratio of the length of the adjacent side to the length of the opposite side in a right triangle. Like the cosecant function, cotangent is also undefined for angles where the sine function equals zero. Similarly, when squared, it is expressed as:

• \( \cot^2 x = \left( \frac{\cos x}{\sin x} \right)^2 = \frac{\cos^2 x}{\sin^2 x} \)

The cotangent function is particularly useful in solving trigonometric equations and can be transformed using trigonometric identities to help prove various relationships.

Pythagorean Identity

The Pythagorean identity is one of the most fundamental identities in trigonometry. It is based on the Pythagorean theorem and relates the square of sine and cosine functions. The basic form is:

• \( \sin^2 x + \cos^2 x = 1 \)

This identity is universally true for any angle \( x \), and it can be manipulated into numerous other trigonometric identities. For instance:

• Expressing \( 1 \) as \( \sin^2 x + \cos^2 x \) allows transformations like \( 1 = \frac{\sin^2 x + \cos^2 x}{\sin^2 x} \) used in algebraic manipulations.
• Solving identities by rewriting expressions to match specific forms, as seen when comparing two sides of a trigonometric equation.

The Pythagorean identity is indispensable when simplifying trigonometric expressions, verifying identities, and proving complex trigonometric relationships across various problems.