Problem 79 - Algebra and Trigonometry

Question

Verify the identity. $$ (\tan x+\cot x)^{2}=\sec ^{2} x+\csc ^{2} x $$

Step-by-Step Solution

Answer

The identity is verified as both sides simplify to \(\sec^2 x + \csc^2 x\).

Step 1: Expand the Left Side

Expand \((\tan x + \cot x)^{2}\) using the formula for the square of a sum \((a+b)^2 = a^2 + 2ab + b^2\). \[(\tan x + \cot x)^2 = \tan^2 x + 2\tan x \cot x + \cot^2 x\]

Step 2: Simplify the Middle Term

Recognize the product \(\tan x \cot x\) as one. Since \(\tan x = \frac{\sin x}{\cos x}\) and \(\cot x = \frac{\cos x}{\sin x}\), multiplying gives \(\tan x \cot x = 1\). Thus, \[2\tan x \cot x = 2\cdot 1 = 2\]

Step 3: Substitute Identities

Substitute the identities \(\tan^2 x = \sec^2 x - 1\) and \(\cot^2 x = \csc^2 x - 1\) into the expanded expression. \[\tan^2 x + 2 + \cot^2 x = (\sec^2 x - 1) + 2 + (\csc^2 x - 1)\]

Step 4: Simplify the Right Side

Combine like terms in the expression \((\sec^2 x - 1) + 2 + (\csc^2 x - 1)\). \[\sec^2 x - 1 + 2 + \csc^2 x - 1 = \sec^2 x + \csc^2 x\]

Step 5: Verify Equality

Now both sides are equal, confirming that the original equation \((\tan x + \cot x)^2 = \sec^2 x + \csc^2 x\) holds true after simplification.

Key Concepts

Trigonometric Functions

Trigonometric functions are essential for understanding angles and their relationships in a triangle, as well as many other areas of mathematics. The primary trigonometric functions are:

• Sine (\(\sin x\)): Ratio of the opposite side to the hypotenuse.
• Cosine (\(\cos x\)): Ratio of the adjacent side to the hypotenuse.
• Tangent (\(\tan x\)): Ratio of the opposite side to the adjacent side, which is also \(\frac{\sin x}{\cos x}\).
• Cotangent (\(\cot x\)): Reciprocal of the tangent, or \(\frac{\cos x}{\sin x}\).
• Secant (\(\sec x\)): Reciprocal of the cosine, or \(\frac{1}{\cos x}\).
• Cosecant (\(\csc x\)): Reciprocal of the sine, or \(\frac{1}{\sin x}\).

Understanding these functions is crucial as they form the basis for more complex functions and identities used in trigonometry. They are also used in calculating angles and distances in various fields, including physics, engineering, and architecture.

Pythagorean Identities

Pythagorean identities are fundamental in trigonometry. They come from the Pythagorean Theorem applied to trigonometric functions. The major Pythagorean identities are:

• \(\sin^2 x + \cos^2 x = 1\), which relates sine and cosine functions.
• \(\tan^2 x + 1 = \sec^2 x\), connecting tangent and secant functions.
• \(1 + \cot^2 x = \csc^2 x\), linking cotangent and cosecant functions.

These identities help in transforming equations and verifying identities, like in the exercise where the identity \((\tan x + \cot x)^2 = \sec^2 x + \csc^2 x\) is verified using these relationships. By knowing these identities, you can simplify trigonometric expressions and solve equations more easily, which is crucial for deeper mathematical analysis.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying expressions to make them easier to work with or to prove a particular property holds true. In trigonometry, this often involves expressing functions in terms of others using identities or converting complex expressions into simpler ones.For example, when verifying trigonometric identities, we might need to apply algebraic manipulation to:

• Expand expressions using identities, like \((a+b)^2 = a^2 + 2ab + b^2\).
• Recognize and simplify terms, such as \(\tan x \cot x = 1\) since they are reciprocals.
• Substitute equivalent trigonometric identities to transform and simplify equations, such as replacing \(\tan^2 x\) with \(\sec^2 x - 1\).

Through algebraic manipulation, we bridge the complex to the simple and enhance our understanding of the relationships between different trigonometric functions. This allows us to explore deeper problems, not only simplifying but also proving and solving equations effectively.