Problem 24 - Algebra and Trigonometry with Analytic Geometry

Question

Verify the identity. $$ \cot 2 u=\frac{\cot ^{2} u-1}{2 \cot u} $$

Step-by-Step Solution

Answer

The identity is verified using trigonometric identities and simplification.

Step 1: Recall Trigonometric Identities

The identity we need to verify is \( \cot(2u) = \frac{\cot^2(u) - 1}{2\cot(u)} \). To verify, we'll use the double angle formula for cotangent: \( \cot(2u) = \frac{\cot^2(u) - 1}{2\cot(u)} \) directly as identity requires no further structural transformation beyond simplifications.

Step 2: Simplifying the Right Side

The right-hand side of the given identity is \( \frac{\cot^2(u) - 1}{2\cot(u)} \). Recall that \( \cot(u) = \frac{\cos(u)}{\sin(u)} \) and \( \cot^2(u) = \frac{\cos^2(u)}{\sin^2(u)} \). We now rewrite \( \cot^2(u) - 1 \) as \( \frac{\cos^2(u) - \sin^2(u)}{\sin^2(u)} \). The expression become the right side: \( \frac{(\cos^2(u) - \sin^2(u))}{2 \cos(u)/\sin(u)} \).

Step 3: Simplifying the Numerator

Notice the identity \( \cos^2(u) - \sin^2(u) = \cos(2u) \). Therefore, insert this into \( \frac{\cos^2(u) - \sin^2(u)}{\sin^2(u)} = \frac{\cos(2u)}{\sin^2(u)} \).

Step 4: Simplifying to Match Cotangent

Now simplify the entire expression: \( \frac{\cos(2u)}{\sin^2(u)} \) becomes \( \frac{\cos(2u) \cdot \sin(u)}{\cos(u)} \) since the denominator itself involves simplification with \( 2 \times \cot(u)= 2 \cdot \sin(u)/\cos(u) \)

Step 5: Rewriting with Double Angle

Finally, using double angle identity of sine \( \sin(2u) = \sin(u) \cos(u) \), confirm that indeed \( \frac{\cos(2u)/\sin(2u)}{\sin^2(u)} = \frac{\sin(u)}{\cos(u)} \cdot \cos(2u)/\cos(u) \). Hence, verified as cotangent expression \( \cot(2u) \).

Key Concepts

cotangent

The cotangent function, often denoted as \( \cot(u) \), is one of the six fundamental trigonometric functions. Its definition is the reciprocal of the tangent function, which means \( \cot(u) = \frac{1}{\tan(u)} \). Mathematically, it's expressed as \( \cot(u) = \frac{\cos(u)}{\sin(u)} \).

Understanding cotangent is crucial in solving problems involving trigonometric identities and expressions:

• The cotangent is undefined wherever the sine of the angle is zero (since division by zero is undefined).
• Cotangent is often used in calculus and analysis to simplify expressions or solve equations.
• It's periodic, meaning it exhibits regular repeating values as the angle \( u \) increases.
  
  

Recognizing and manipulating cotangent expressions can help in verifying trigonometric identities, as it often appears in simplified forms.

double angle formula

The double angle formulas are specific trigonometric identities used to express functions of double angles, \( 2u \), in terms of single angles, \( u \). These formulas are highly useful in calculus and algebra for simplifying expressions.

For cotangent, the double angle formula is stated as:

• \( \cot(2u) = \frac{\cot^2(u) - 1}{2\cot(u)} \)

This formula helps convert complex expressions into simpler ones by leveraging powers of cotangent.

Understanding double angle formulas is essential when manipulating and verifying trigonometric identities:

• They simplify integrations and derivations involving trigonometric functions.
• They are used extensively in both pure and applied mathematics, including engineering and physics.
• Memorizing these formulas allows for quick and efficient problem-solving.
  

Practicing problems that involve these formulas can deepen your understanding and improve your problem-solving skills.

simplification of trigonometric expressions

Simplifying trigonometric expressions involves rewriting them in a more easily understandable or computable form. This process often uses fundamental identities such as Pythagorean, reciprocal, and double angle identities.

To simplify an expression:

• Identify the trigonometric identity that can be applied.
• Use algebraic techniques like factoring or combining like terms.
• Replace complex power forms with simpler equivalent expressions.
  
  

For example, given a complex expression involving \( \cot^2(u) - 1 \), you can use the identity \( \cot(2u) = \frac{\cot^2(u) - 1}{2\cot(u)} \) to simplify the expression into a form that’s easier to handle or solve.

This process is vital for verifying trigonometric identities and solving complex mathematical problems, making it a fundamental skill in trigonometry.

verifying trigonometric identities

Verifying trigonometric identities involves demonstrating that two expressions are equivalent under specific conditions. It's not unlike proving a theorem in mathematics and requires a step-by-step logical approach.

To verify a trigonometric identity, consider these steps:

• Start with one side of the equation, usually the more complex one, and manipulate it to look like the other side.
• Employ known trigonometric identities strategically (such as double angle formulas).
• Simplify the expression step by step, ensuring each step is mathematically correct.
• Match the simplified form of the left-hand side with the right-hand side to complete the verification.
  

This requires a strong understanding of various trigonometric identities and algebraic manipulation.

Successful verification not only validates the identity but also enhances your comprehension of underlying trigonometric principles and their interconnections across different mathematical domains.