Problem 81 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

Identity is verified since both sides simplify to \( \sec^2 x + \csc^2 x \).

Step 1: Simplify Left Side

To verify the identity, start by expanding the left side: \[ ( an x + rac{1}{ an x})^2 \]This becomes:\[ ( an^2 x + 2 + rac{1}{ an^2 x}) \]

Step 2: Convert to Sec and Csc

Recognize that \( an^2 x = rac{ ext{sec}^2 x - 1}{1} \) and \( rac{1}{ an^2 x} = ext{csc}^2 x - 1 \). Substitute these identities into the expanded expression:\[ rac{ ext{sec}^2 x - 1}{1} + 2 + ( ext{csc}^2 x - 1) \]

Step 3: Simplify the Expression

Combine like terms in the equation from Step 2:\[ ext{sec}^2 x - 1 + 2 + ext{csc}^2 x - 1 \]This simplifies to:\[ ext{sec}^2 x + ext{csc}^2 x \]

Step 4: Compare with Right Side

Now, compare the simplified expression from Step 3 with the right side of the original identity. The expression \( ext{sec}^2 x + ext{csc}^2 x \) matches the right-hand side of the identity.

Key Concepts

Tangent and Cotangent: Understanding Their Role and Relationship

Tangent and cotangent are two fundamental trigonometric functions that have a unique reciprocal relationship. Tangent, often denoted as \( \tan x \), is the ratio of the opposite side to the adjacent side in a right-angled triangle. Mathematically, it can be expressed as:\[ \tan x = \frac{\sin x}{\cos x} \]Cotangent, on the other hand, is the reciprocal of tangent. It is denoted as \( \cot x \) and can be defined as the ratio of the adjacent side to the opposite side, or:\[ \cot x = \frac{1}{\tan x} = \frac{\cos x}{\sin x} \]
Understanding these relationships is crucial, especially when verifying identities like \((\tan x + \cot x)^{2}\). This expression is simplified by recognizing that both \( \tan x \) and \( \cot x \) can be represented in terms of sine and cosine. This helps in rewriting complex trigonometric expressions and aids in the process of simplification.

Secant and Cosecant: Leveraging Their Forms in Trigonometric Identities

Secant and cosecant are also important trigonometric functions that frequently appear in various identities. Secant, denoted as \( \sec x \), is the reciprocal of cosine, making it:\[ \sec x = \frac{1}{\cos x} \]Similarly, cosecant, denoted as \( \csc x \), is the reciprocal of sine:\[ \csc x = \frac{1}{\sin x} \]
In trigonometric identities, these reciprocal functions help transform expressions by replacing terms like \( \tan^2 x \) and \( \cot^2 x \) with their reciprocal equivalents. For instance, recognizing that \( \tan^2 x = \sec^2 x - 1 \) and \( \cot^2 x = \csc^2 x - 1 \) is critical. This understanding helps in converting and simplifying complex expressions, such as when proving the identity \((\tan x + \cot x)^{2} = \sec^2 x + \csc^2 x\).

Simplifying Expressions: A Step-by-Step Approach to Trigonometric Identities

One of the essential skills in mastering trigonometry is simplifying expressions by using identities. When you encounter a trigonometric identity like \((\tan x + \cot x)^{2} = \sec^2 x + \csc^2 x\), the first step is to transform the left side to an easily recognizable form.Begin by expanding \((\tan x + \cot x)^{2}\), which translates into \( \tan^2 x + 2\tan x \cot x + \cot^2 x \). Since \( \tan x \cot x = 1 \), you simplify it to \( \tan^2 x + \cot^2 x + 2 \).Next, leverage the identities such as \( \tan^2 x = \sec^2 x - 1 \) and \( \cot^2 x = \csc^2 x - 1 \). By substituting these, you simplify the expression into the form \( \sec^2 x + \csc^2 x \), matching the right side of the identity.These simplification techniques help in recognizing patterns and relationships among trigonometric functions. The correct use of identities and substitution plays a vital role in proving and understanding complex trigonometric equations.