Problem 87 - Algebra and Trigonometry

Question

Verify the identity. $$ (\tan x+\cot x)^{4}=\csc ^{4} x \sec ^{4} x $$

Step-by-Step Solution

Answer

The identity is verified; both sides simplify to \( \frac{1}{\sin^4 x \cos^4 x} \).

Step 1: Express Trigonometric Functions in Terms of Sine and Cosine

Start by expressing the tangent and cotangent in terms of sine and cosine. We know that \( \tan x = \frac{\sin x}{\cos x} \) and \( \cot x = \frac{\cos x}{\sin x} \). Therefore, \( \tan x + \cot x = \frac{\sin x}{\cos x} + \frac{\cos x}{\sin x} \).

Step 2: Simplify the Expression

Combine the fractions from the previous step. The common denominator is \( \sin x \cos x \), so:\[\tan x + \cot x = \frac{\sin x}{\cos x} + \frac{\cos x}{\sin x} = \frac{\sin^2 x + \cos^2 x}{\sin x \cos x}\]Since \( \sin^2 x + \cos^2 x = 1 \), the expression simplifies to:\[\tan x + \cot x = \frac{1}{\sin x \cos x}\]

Step 3: Raise the Expression to the Fourth Power

Now, raise both sides of the simplified equation to the fourth power:\[(\tan x + \cot x)^4 = \left(\frac{1}{\sin x \cos x}\right)^4 = \frac{1}{(\sin x \cos x)^4}\]

Step 4: Simplify the Right Side of the Identity

Express the right side of the given identity \( \csc^4 x \sec^4 x \) in terms of sine and cosine. Using \( \csc x = \frac{1}{\sin x} \) and \( \sec x = \frac{1}{\cos x} \), we can write:\[\csc^4 x \sec^4 x = \left(\frac{1}{\sin x}\right)^4 \left(\frac{1}{\cos x}\right)^4 = \frac{1}{\sin^4 x \cos^4 x}\]

Step 5: Verify the Identity

Compare the results from Steps 3 and 4. Both expressions \( \frac{1}{(\sin x \cos x)^4} \) and \( \frac{1}{\sin^4 x \cos^4 x} \) are identical. Hence, the identity is verified.

Key Concepts

Tangent and Cotangent

In trigonometry, tangent and cotangent are fundamental concepts that help describe angles and triangles. Tangent (\( \tan x \) ) is defined as the ratio of the sine of an angle to the cosine of the angle, represented as:

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

Cotangent (\( \cot x \) ) is the reciprocal of tangent, or the ratio of the cosine of an angle to the sine, given by:

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

These ratios help simplify and solve trigonometric identities and equations.
For example, expressing tangent and cotangent in terms of sine and cosine allowed us to simplify the expression in the exercise.The sum of tangent and cotangent \( \left( \tan x + \cot x \right) \) was simplified using a common denominator, indicating the interplay between these trigonometric functions.

Sine and Cosine

Sine and cosine are the cornerstone of trigonometry, forming the basis for understanding angles and their spatial relationships. Sine (\( \sin x \) ) represents the opposite side over the hypotenuse in a right triangle, while cosine (\( \cos x \) ) is the adjacent side over the hypotenuse. These functions are crucial for finding relationships between angles and sides.
Their fundamental property is expressed in the Pythagorean identity:

• \( \sin^2 x + \cos^2 x = 1 \)

This identity is pivotal for solving many trigonometric problems.
In the given solution, this identity simplified the expression of \( \tan x + \cot x \) to \( \frac{1}{\sin x \cos x} \), emphasizing the power of sine and cosine to reduce complexity and verify identities.

Cosecant and Secant

Cosecant and secant are derived from sine and cosine and serve as their respective reciprocals. They further expand the understanding of trigonometric functions.

• The cosecant, denoted as \( \csc x \), is the reciprocal of sine: \( \csc x = \frac{1}{\sin x} \).
• The secant, abbreviated as \( \sec x \), is the reciprocal of cosine: \( \sec x = \frac{1}{\cos x} \).

These functions are especially useful in identities and equations that require division by sine or cosine.
For instance, in the exercise, expressing \( \csc^4 x \sec^4 x \) using sine and cosine simplified the verification process. Using these reciprocal relationships allowed for matching both sides of the identity by rewriting each side in terms of sine and cosine, thus confirming their equality.