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Chapter 5: Problem 14

Prove that each of the following identities is true: $$ \sin ^{2} x\left(\cot ^{2} x+1\right)=1 $$

Short Answer

Expert verified
The identity is true because \( \sin^2 x (\cot^2 x + 1) \) simplifies to 1 using trigonometric identities.

Step by step solution

01

Recall Trigonometric Identities

To solve this problem, recall two essential trigonometric identities:1. The Pythagorean identity: \[ \sin^2 x + \cos^2 x = 1 \]2. The identity for cotangent in terms of sine and cosine: \[ \cot x = \frac{\cos x}{\sin x} \] Knowing these will help in transforming the left-hand side of the identity.
02

Express Cotangent Squared in Terms of Sine and Cosine

We have \( \cot^2 x = \left( \frac{\cos x}{\sin x} \right)^2 = \frac{\cos^2 x}{\sin^2 x} \). Using this, we can rewrite \( \cot^2 x + 1 \) as:\[ \cot^2 x + 1 = \frac{\cos^2 x}{\sin^2 x} + 1 \].
03

Simplify Expression Using Common Denominator

The expression \( \frac{\cos^2 x}{\sin^2 x} + 1 \) can be rewritten with a common denominator:\[ \frac{\cos^2 x}{\sin^2 x} + \frac{\sin^2 x}{\sin^2 x} = \frac{\cos^2 x + \sin^2 x}{\sin^2 x} \].Using the Pythagorean identity, \( \cos^2 x + \sin^2 x = 1 \), simplifies this to:\[ \frac{1}{\sin^2 x} \].
04

Substitute into the Original Expression

Substitute \( \frac{1}{\sin^2 x} \) from Step 3 into the left-hand side of the original equation:\[ \sin^2 x \cdot \left( \cot^2 x + 1 \right) = \sin^2 x \cdot \frac{1}{\sin^2 x} \].
05

Simplify the Expression

The multiplication \( \sin^2 x \cdot \frac{1}{\sin^2 x} \) simplifies to 1, since:\[ \sin^2 x \cdot \frac{1}{\sin^2 x} = 1 \].Thus, the identity \( \sin^2 x (\cot^2 x + 1) = 1 \) is verified.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Pythagorean Identity
The Pythagorean identity is a fundamental concept in trigonometry. It states that for any angle \( x \), the square of the sine of \( x \) plus the square of the cosine of \( x \) equals one. In mathematical terms, it is written as \( \sin^2 x + \cos^2 x = 1 \).

This identity is extremely useful because it relates sine and cosine, which are the primary trigonometric functions. Essentially, this means that in a right-angled triangle, the sum of the squares of the sides adjacent to the hypotenuse equals the square of the hypotenuse itself.

Here are some key points about the Pythagorean identity:
  • It helps in transforming trigonometric expressions, as seen in solving the given identity.
  • This identity is derived from the Pythagorean theorem in geometry, applied to the unit circle.
  • Many other trigonometric identities are derived from this basic identity.
Cotangent Function
The cotangent function, denoted as \( \cot x \), is one of the six fundamental trigonometric functions. It is the reciprocal of the tangent function. In terms of sine and cosine, cotangent can be expressed as \( \cot x = \frac{\cos x}{\sin x} \).

Understanding the cotangent function involves knowing how it compares to the tangent function. While \( \tan x = \frac{\sin x}{\cos x} \), \( \cot x \) flips this relationship.

Here are some quick facts about the cotangent function:
  • Cotangent is undefined when \( \sin x = 0 \), because division by zero is not possible.
  • The function \( \cot^2 x + 1 \) is often encountered in identities, exemplifying its relationship with other trigonometric terms.
  • In terms of a sine and cosine ratio, it can help convert complex trigonometric expressions into simpler forms.
Identity Verification
Verifying an identity means showing that both sides of a given equation are equivalent, using known identities or algebraic manipulations. For this exercise, the goal was to prove that \( \sin^2 x (\cot^2 x + 1) = 1 \).

The process of verification followed these key steps:
  • Recognize and utilize essential trigonometric identities, such as the Pythagorean identity and the expression for cotangent.
  • Simplify the equation step by step, expressing cotangent squared as \( \frac{\cos^2 x}{\sin^2 x} \) and combining terms using a common denominator.
  • Substitute and simplify until the expression reduces to a basic, undeniable truth or number, like 1 in this case.
Successful verification involves meticulous observation of both sides of an identity, leveraging basic identities, and performing calculations carefully. Through these steps, the provided identity is shown to hold true, demonstrating the logical structure of trigonometric equations.

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Most popular questions from this chapter

Prove that each of the following identities is true: $$ \frac{\sin x+1}{\cos x+\cot x}=\tan x $$

Prove that each of the following identities is true: $$ \csc ^{4} \theta-\cot ^{4} \theta=\frac{1+\cos ^{2} \theta}{\sin ^{2} \theta} $$

Prove each identity. $$ \csc \theta+\sin (-\theta)=\frac{\cos ^{2} \theta}{\sin \theta} $$

Prove that each of the following identities is true: $$ \frac{1-\cos ^{3} A}{1-\cos A}=\cos ^{2} A+\cos A+1 $$

The following identities are from the book Plane and Spherical Trigonometry with Tables by Rosenbach, Whitman, and Moskovitz, and published by Ginn and Company in 1937 . Verify each identity. $$ \frac{1+\sin \phi}{1-\sin \phi}-\frac{1-\sin \phi}{1+\sin \phi}=4 \tan \phi \sec \phi $$
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