Problem 84 - Algebra and Trigonometry

Question

Verify the identity. $$ \frac{\tan v-\cot v}{\tan ^{2} v-\cot ^{2} v}=\sin v \cos v $$

Step-by-Step Solution

Answer

The identity is verified as both sides simplify to \( \sin v \cos v \).

Step 1: Rewrite tan and cot in terms of sine and cosine

Express \( \tan v \) and \( \cot v \) in terms of \( \sin v \) and \( \cos v \). Recall that \( \tan v = \frac{\sin v}{\cos v} \) and \( \cot v = \frac{\cos v}{\sin v} \).

Step 2: Substitute values into the identity

Replace \( \tan v \) and \( \cot v \) in the equation \( \frac{\tan v - \cot v}{\tan^2 v - \cot^2 v} \) with \( \frac{\sin v}{\cos v} - \frac{\cos v}{\sin v} \) in the numerator and \( \left(\frac{\sin v}{\cos v}\right)^2 - \left(\frac{\cos v}{\sin v}\right)^2 \) in the denominator.

Step 3: Simplify the numerator

Combine the terms in the numerator: \( \frac{\sin v}{\cos v} - \frac{\cos v}{\sin v} = \frac{\sin^2 v - \cos^2 v}{\sin v \cos v} \).

Step 4: Simplify the denominator

Apply the difference of squares in the denominator: \( \left(\frac{\sin v}{\cos v}\right)^2 - \left(\frac{\cos v}{\sin v}\right)^2 = \frac{\sin^2 v}{\cos^2 v} - \frac{\cos^2 v}{\sin^2 v} = \frac{(\sin^4 v - \cos^4 v)}{\sin^2 v \cos^2 v} \).

Step 5: Factor the difference of squares in denominator

Express \( \sin^4 v - \cos^4 v \) as a difference of squares: \((\sin^2 v)^2 - (\cos^2 v)^2 = (\sin^2 v - \cos^2 v)(\sin^2 v + \cos^2 v) \). The identity \( \sin^2 v + \cos^2 v = 1 \) simplifies this to \( \sin^2 v - \cos^2 v \).

Step 6: Complete the simplification

Cancel \( \sin^2 v - \cos^2 v \) from the numerator and denominator: \( \frac{\sin^2 v - \cos^2 v}{\sin^2 v - \cos^2 v} \cdot \frac{\sin v \cos v}{\sin^2 v \cos^2 v} = \frac{\sin v \cos v}{\sin^2 v \cos^2 v} = \sin v \cos v \), confirming it equals the right side of the identity.

Key Concepts

Difference of Squares

The concept of the difference of squares is a fundamental algebraic identity that simplifies expressions. When you see a difference like \( a^2 - b^2 \), it can be factored into \((a - b)(a + b)\). This principle helps to break down the expressions into simpler forms for easier manipulation. In the given problem, we see this applied in the step where \( \sin^4 v - \cos^4 v \) is rewritten. It becomes \( (\sin^2 v)^2 - (\cos^2 v)^2 \).

• Breaking it down: \((\sin^2 v - \cos^2 v)(\sin^2 v + \cos^2 v)\)
• Use the identity: \(\sin^2 v + \cos^2 v = 1\)

Thus, simplifying snags in trigonometric identities often involve spotting these differences and factoring them. This paves the way for easier cancellation of terms, as seen in the detailed solution.

Simplifying Algebraic Expressions

Simplifying algebraic expressions involves reducing complex equations into simpler and more interpretable forms. In the context of the problem, simplifying involves a couple of steps, notably in the numerator and denominator of the fraction. First, rewriting \( \tan v \) and \( \cot v \) using sine and cosine to express the terms in a common framework is key. Then, combining and simplifying these terms:

• Numerator: \( \frac{\sin v}{\cos v} - \frac{\cos v}{\sin v} \) simplifies to \( \frac{\sin^2 v - \cos^2 v}{\sin v \cos v} \)
• Denominator: Transform difference of squares \((\frac{\sin^2 v}{\cos^2 v} - \frac{\cos^2 v}{\sin^2 v})\)

Crucially, once simplified, these expressions can often be reduced or canceled if written correctly, demonstrating a clear path to the final identity check \( \sin v \cos v \). Each simplification step leans heavily on algebraic manipulation and the understanding of trigonometric identities.

Trigonometric Functions

Trigonometric functions such as sine, cosine, tangent, and cotangent play a crucial role in forming these identities. Understanding their interrelations is fundamental. In our problem, knowing the identities \( \tan v = \frac{\sin v}{\cos v} \) and \( \cot v = \frac{\cos v}{\sin v} \) is vital, as we use them to express everything in terms of sine and cosine only. Trigonometric functions are interrelated:

• Basic identities like \( \sin^2 v + \cos^2 v = 1\) are key for simplification.
• The manipulation and rewriting of these into each other allows for simplification and verification of complex identities.

When working with identities, recognizing these functions and using their fundamental relationships allows the reduction of any complex expression to its simplest form. Mastery of these relationships opens vast ways to solve equations efficiently, as illustrated in the solution verification of this exercise.