Problem 86 - 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: both sides simplify to \( \sin v \cos v \).

Step 1: Express Tan and Cot

Begin by expressing \( \tan v \) and \( \cot v \) in terms of sine and cosine. We know that \( \tan v = \frac{\sin v}{\cos v} \) and \( \cot v = \frac{\cos v}{\sin v} \). Substitute these expressions into the left-hand side of the identity.

Step 2: Simplify the Numerator

Substitute \( \tan v = \frac{\sin v}{\cos v} \) and \( \cot v = \frac{\cos v}{\sin v} \) into the numerator: \( \tan v - \cot v = \frac{\sin v}{\cos v} - \frac{\cos v}{\sin v} \). Find a common denominator to simplify: \( \frac{\sin^2 v - \cos^2 v}{\sin v \cos v} \).

Step 3: Simplify the Denominator

Now substitute into the denominator: \( \tan^2 v - \cot^2 v = \left(\frac{\sin v}{\cos v}\right)^2 - \left(\frac{\cos v}{\sin v}\right)^2 \). This simplifies to \( \frac{\sin^2 v}{\cos^2 v} - \frac{\cos^2 v}{\sin^2 v} \).Applying the common denominator, it becomes:\( \frac{(\sin^2 v)^2 - (\cos^2 v)^2}{\sin^2 v \cos^2 v} \).

Step 4: Apply the Difference of Squares

Notice that both the simplified numerator and denominator expressions can be written using the difference of squares:For the numerator, \( \sin^2 v - \cos^2 v = (\sin v - \cos v)(\sin v + \cos v) \).For the denominator, \( (\sin^2 v)^2 - (\cos^2 v)^2 = (\sin^2 v - \cos^2 v)(\sin^2 v + \cos^2 v) = (\sin^2 v - \cos^2 v)(1) \) because \(\sin^2 v + \cos^2 v = 1\).

Step 5: Divide and Simplify

Now divide the simplified numerator by the simplified denominator:\[ \frac{\frac{\sin^2 v - \cos^2 v}{\sin v \cos v}}{\frac{\sin^2 v - \cos^2 v}{\sin^2 v \cos^2 v}} = \frac{(\sin v - \cos v)(\sin v + \cos v)}{(\sin^2 v - \cos^2 v)} \cdot \sin v \cos v. \]The \( \sin^2 v - \cos^2 v \) cancels out, leaving: \( \sin v \cos v \).

Step 6: Validate the Identity

After simplification, we have shown that both sides of the equation are equal, hence:\[ \frac{\tan v - \cot v}{\tan^2 v - \cot^2 v} = \sin v \cos v \].

Step 7: Conclusion: Identity Verified

The identity \( \frac{\tan v - \cot v}{\tan^2 v - \cot^2 v} = \sin v \cos v \) is verified as true through simplification and using trigonometric identities.

Key Concepts

Simplifying Trigonometric Expressions

Simplifying trigonometric expressions is a crucial skill in trigonometry. It involves rewriting complex trigonometric expressions into simpler forms using known identities and algebraic techniques.

In the exercise, we start by substituting

• \( \tan v = \frac{\sin v}{\cos v} \)
• \( \cot v = \frac{\cos v}{\sin v} \)

into the expression. We aim to rewrite it in terms of sine and cosine, which are potentially simpler to work with. This transformation makes the structure of the trigonometric identity more apparent.

Finding a common denominator in the process helps in merging separate fractions, simplifying them further. In the given exercise, transforming both the numerator and denominator individually is necessary to see the resemblance to the known trigonometric identities. Thus, simplifying effectively requires

• Identifying common trigonometric identities.
• Applying algebraic manipulation techniques such as finding common denominators and combining fractions.

Mastering these simplification techniques opens the doorway to solving more complex trigonometric equations and verifying identities, as shown in the given example.

Difference of Squares

The difference of squares is a powerful algebraic technique used within trigonometry. It relies on the identity

• \( a^2 - b^2 = (a-b)(a+b) \).

In trigonometry, this method is seamlessly applied to expressions that can be reshaped into a difference of square forms. In the step-by-step solution, this technique is applied as follows:

• For the numerator: \( \sin^2 v - \cos^2 v = (\sin v - \cos v)(\sin v + \cos v) \).
• For the denominator: \( (\sin^2 v)^2 - (\cos^2 v)^2 = (\sin^2 v - \cos^2 v)(1) \), given \( \sin^2 v + \cos^2 v = 1 \).

The recognition of these patterns simplifies the manipulation of complex expressions. It allows us to factor, cancel terms, and simplify the remaining components more effectively. Recognizing and applying the difference of squares transforms an otherwise difficult problem into a more straightforward computation, making trigonometric simplifications much more manageable.

Sine and Cosine

Sine and cosine are fundamental trigonometric functions every student should be familiar with. They are defined as:

• Sine \( (\sin \theta) \): opposite side over hypotenuse in a right triangle.
• Cosine \( (\cos \theta) \): adjacent side over hypotenuse in a right triangle.

In the realm of identities and simplifications, they play pivotal roles as they form the basis of many trigonometric identities such as

• \( \tan \theta = \frac{\sin \theta}{\cos \theta} \)
• \( \cot \theta = \frac{\cos \theta}{\sin \theta} \)

In the exercise, expressing \( \tan v \) and \( \cot v \) in terms of sine and cosine helps to achieve the desired simplification. They not only simplify computations but also unveil underlying relationships like \( \sin^2 v + \cos^2 v = 1 \), which is often used in various trigonometric simplifications and proofs.

Understanding how sine and cosine transform and interact is essential for effectively solving and verifying trigonometric identities, as illustrated in the exercise.