Problem 34 - Algebra and Trigonometry

Question

Verify the identity. $$ \csc x[\csc x+\sin (-x)]=\cot ^{2} x $$

Step-by-Step Solution

Answer

The given identity is verified by simplifying both sides to the same expression.

Step 1: Rewrite using trigonometric identities

First, recall the trigonometric identities for cosecant and sine: \( \csc x = \frac{1}{\sin x} \) and \( \sin(-x) = -\sin x \). Substitute these into the left side of the equation.

Step 2: Substitute identities into the equation

Replace \( \csc x \) and \( \sin(-x) \) in the original expression. The left-hand side becomes \( \csc x \left( \csc x + (-\sin x) \right) \), which simplifies to \( \csc x (\csc x - \sin x) \).

Step 3: Simplify the expression

Since \( \csc x = \frac{1}{\sin x} \), substitute \( \csc x \) accordingly: \( \left( \frac{1}{\sin x} \right)\left( \frac{1}{\sin x} - \sin x \right) \). This simplifies to \( \frac{1}{\sin^2 x} - \frac{\sin x}{\sin x} \).

Step 4: Simplify the fraction

\( \frac{\sin x}{\sin x} = 1 \). Thus, the expression becomes \( \frac{1}{\sin^2 x} - 1 \).

Step 5: Use Pythagorean Identity

The Pythagorean identity \( \csc^2 x - 1 = \cot^2 x \) can be used here. Substitute \( \csc^2 x = \frac{1}{\sin^2 x} \) into \( \csc^2 x - 1 \), confirming it equals \( \cot^2 x \). The left-hand side \( \frac{1}{\sin^2 x} - 1 \) now equals \( \cot^2 x \), thus verifying the identity.

Key Concepts

Cosecant Function

The cosecant function is one of the reciprocal trigonometric functions. If you're familiar with the sine function, then cosecant is its reciprocal. Mathematically, the identity is expressed as:

• \( ext{Cosecant of } x: \, \csc x = \frac{1}{\sin x} \)

This implies that wherever the sine function is defined, the cosecant function also holds value except where sine is zero. This is because division by zero is undefined in mathematics.
To visualize it, imagine the unit circle. At certain angles where sine equals zero, such as at 0 degrees or 180 degrees, the cosecant function spikes towards infinity. These characteristics make the cosecant function particularly useful in identities involving reciprocal relationships in trigonometry.

Sine Function

The sine function is a fundamental trigonometric function. It's one of the basic building blocks of trigonometric identities and is used widely in mathematics and engineering. Its identity is defined as the ratio between the opposite side and the hypotenuse in a right triangle:

• \( ext{Sine of } x: \, \sin x = \frac{\text{opposite}}{\text{hypotenuse}} \)

An important aspect of the sine function is its periodicity. It repeats values every 360 degrees or 2\(\pi\) radians, making it useful to model repeating phenomena such as waves.
Additionally, functions like sine possess symmetry properties. For instance, \(\sin(-x) = -\sin x\), highlighting its odd function nature, which was essential in simplifying the given expression by turning \(\sin(-x)\) into \(-\sin x\). So, understanding sine's properties not only aids in simplifying trigonometric expressions but is also essential to grasping more complex identities.

Pythagorean Identity

The Pythagorean identity is a fundamental concept in trigonometry linking sine and cosine. Though, in this context, it is used to connect cosecant and cotangent functions. The standard Pythagorean identity is:

• \( \sin^2 x + \cos^2 x = 1 \)

Through manipulation, this identity leads to other relationships:

• \( 1 + \cot^2 x = \csc^2 x \)

This particular form was vital in our solution. It was used to transform the expression \( \frac{1}{\sin^2 x} - 1 \) into \( \cot^2 x \).
By remembering this derived form of the Pythagorean identity, it becomes easier to verify and simplify many trigonometric proofs and problems. Recognizing these equivalences can simplify solving equations and verifying identities significantly.

Cotangent Function

The cotangent function is another reciprocal trigonometric function, the counterpart of tangent. Defined mathematically, it is the reciprocal of the tangent function:

• \( ext{Cotangent of } x: \, \cot x = \frac{\cos x}{\sin x} \)

Cotangent helps in evaluating and relating angles in trigonometry, particularly in problems that involve perpendicular lines.
In our identity verification, the expression for cotangent squared, \( \cot^2 x \), was key. It stems from the identity \( \csc^2 x - 1 = \cot^2 x \), efficiently turning a trigonometric expression into more manageable terms.
Understanding cotangent's relationship to both sine and cosine allows for greater flexibility in managing complex trigonometric functions. Moreover, familiarizing yourself with cotangent can often reveal the symmetry or patterns hidden in trigonometric calculations.