Problem 70 - Precalculus Mathematics for Calculus

Question

Verify the identity. $$\frac{\csc ^{2} x-\cot ^{2} x}{\sec ^{2} x}=\cos ^{2} x$$

Step-by-Step Solution

Answer

The identity \( \frac{\csc^2 x - \cot^2 x}{\sec^2 x} = \cos^2 x \) is verified.

Step 1: Rewrite using Trigonometric Identities

We start by rewriting the trigonometric functions using their identities: 1. \( \csc^2 x = \frac{1}{\sin^2 x} \) and \( \cot^2 x = \frac{\cos^2 x}{\sin^2 x} \) can be used to rewrite the numerator:\[ \csc^2 x - \cot^2 x = \frac{1}{\sin^2 x} - \frac{\cos^2 x}{\sin^2 x}. \]2. \( \sec^2 x = \frac{1}{\cos^2 x} \) applies to the denominator.

Step 2: Simplify the Numerator

Combine the terms in the numerator:\[ \frac{1}{\sin^2 x} - \frac{\cos^2 x}{\sin^2 x} = \frac{1 - \cos^2 x}{\sin^2 x}. \]Use the Pythagorean identity, \( 1 - \cos^2 x = \sin^2 x \), to simplify further:\[ \frac{\sin^2 x}{\sin^2 x} = 1. \]

Step 3: Substitute and Simplify the Entire Expression

Now, substitute back into the original identity:\[ \frac{\csc^2 x - \cot^2 x}{\sec^2 x} = \frac{1}{\frac{1}{\cos^2 x}}. \]Simplify the fraction on the right:\[ \cos^2 x. \]

Step 4: Conclusion

Both sides of the original equation are now equal:\[ \cos^2 x = \cos^2 x. \]Thus, the identity is verified.

Key Concepts

Cosecant

The cosecant, often abbreviated as \( \csc \), is a fundamental trigonometric function. It is the reciprocal of the sine function. If you have a right triangle, the sine of an angle \( x \) is the ratio of the length of the opposite side to the hypotenuse. The cosecant of \( x \) is the inverse of this ratio, and can be expressed as:

• \( \csc x = \frac{1}{\sin x} \)

This relationship shows that the cosecant function is undefined wherever the sine is zero, since division by zero is not allowed in mathematics.
For an angle \( x \) in the unit circle, where the radius is 1, the cosecant still holds as \( \csc x = \frac{1}{y} \) when \( y \) is the sine of the angle. This concept is critical in understanding how trigonometric identities interrelate, particularly in transformations and manipulations found in algebraic proofs and verifications like the one presented in the exercise above.
Using the identity \( \csc^2 x = \frac{1}{\sin^2 x} \), it becomes possible to manipulate and simplify complex trigonometric expressions.

Secant

The secant, denoted as \( \sec \), is another reciprocal trigonometric function like the cosecant. However, it relates to the cosine function. The cosine of an angle \( x \) in a right triangle is the ratio of the adjacent side to the hypotenuse. The secant is the reciprocal of this ratio:

• \( \sec x = \frac{1}{\cos x} \)

Like the cosecant, the secant is undefined at points where the cosine equals zero, leading to asymptotic behavior in its graph.
In the context of a unit circle where the radius is one, the secant function can also be defined as \( \sec x = \frac{1}{x} \) when the \( x \) coordinate represents the cosine of the angle. This definition helps in transforming and graphing the function, as well as in deriving its properties via trigonometric identities.
In trigonometric identities, \( \sec^2 x = \frac{1}{\cos^2 x} \) plays a crucial role in proofs and simplifications, much like how it was used in the original exercise to verify the identity.

Cotangent

The cotangent, represented by \( \cot \), is the reciprocal of the tangent function, which is another primary trigonometric ratio. In a right triangle, tangent relates the opposite and adjacent sides of an angle. The cotangent therefore is:

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

Given this relationship, the cotangent is undefined for angles where the tangent is zero (i.e., when the sine is zero because division by sine is involved).
When considered on the unit circle, the properties of cotangent make it a useful tool for evaluating angles and solving trigonometric equations, allowing conversion between different forms.
In verifying identities, as shown in the exercise, \( \cot^2 x \) can be rewritten using identities like \( \cot^2 x = \frac{\cos^2 x}{\sin^2 x} \), aiding in algebraic transformations to reduce complex expressions into simpler or more useful forms for comparison or simplification.
This approach helps ensure consistency and accuracy when working with the wide variety of trigonometric equations.