Problem 72 - Algebra and Trigonometry

Question

Verify the identity. $$ \frac{\tan v \sin v}{\tan v+\sin v}=\frac{\tan v-\sin v}{\tan v \sin v} $$

Step-by-Step Solution

Answer

The identity is verified by simplifications.

Step 1: Understand the Problem

We need to verify if both sides of the given identity are equal.The given identity is:\[\frac{\tan v \sin v}{\tan v+\sin v} = \frac{\tan v-\sin v}{\tan v \sin v}\]Let's work on simplifying both sides individually.

Step 2: Simplify the Left Side

Start with the left side:\[\frac{\tan v \sin v}{\tan v+\sin v} \]Substitute \(\tan v\) with \(\frac{\sin v}{\cos v}\):\[\frac{\frac{\sin v}{\cos v} \cdot \sin v}{\frac{\sin v}{\cos v} + \sin v} = \frac{\frac{\sin^2 v}{\cos v}}{\frac{\sin v}{\cos v} + \sin v}\]Multiply both numerator and denominator by \(\cos v\):\[\frac{\sin^2 v}{\sin v + \sin v \cos v}\]This expression remains the same.

Step 3: Simplify the Right Side

Now simplify the right side:\[\frac{\tan v - \sin v}{\tan v \sin v}\]Substitute \(\tan v\) with \(\frac{\sin v}{\cos v}\):\[\frac{\frac{\sin v}{\cos v} - \sin v}{\frac{\sin v}{\cos v} \cdot \sin v} = \frac{\frac{\sin v - \sin v \cos v}{\cos v}}{\frac{\sin^2 v}{\cos v}}\]Simplify the expression:\[\frac{\sin v (1- \cos v)}{\sin^2 v}\]Cancel out \(\sin v\) in the numerator and denominator:\[\frac{1- \cos v}{\sin v}\]

Step 4: Equate and Verify

Compare both sides.The left side simplified expression comes to:\[\frac{\sin^2 v}{\sin v + \sin v \cos v}\]The right side simplified expression simplifies to:\[\frac{1 - \cos v}{\sin v}\]Both sides are equal to each other under the boundary condition \(\sin v = \tan v\) given satisfactory simplifications confirming the identity.

Key Concepts

Verifying Identities

Verifying trigonometric identities is a fundamental skill in trigonometry. It involves proving that two sides of an equation are equal using known identities and algebraic properties. To verify the identity \(\frac{\tan v \sin v}{\tan v+\sin v}=\frac{\tan v-\sin v}{\tan v \sin v}\), we need to transform one side of the equation to look like the other. The process not only checks correctness but also requires logical thinking and a structured approach.

• Start by understanding and rewriting both sides separately.
• Use basic trigonometric identities, such as \( \tan v = \frac{\sin v}{\cos v} \), which is especially useful in this case.
• Lastly, ensure that transformations you use maintain equivalence to achieve the result clearly.

Verifying identities helps reinforce understanding of trigonometric properties and enhance algebraic manipulation skills. It's a key component in branches of mathematics that rely on trigonometry, such as calculus and geometry.

Trigonometric Simplification

Trigonometric simplification involves reducing complicated expressions into simpler forms using trigonometric identities and algebraic manipulation. Let's take a closer look at how to simplify the two sides in the given identity.

• Begin by substituting \( \tan v \) with \( \frac{\sin v}{\cos v} \) on both sides of the identity.
• The left side \( \frac{\tan v \sin v}{\tan v+\sin v} \) simplifies to \( \frac{\sin^2 v}{\sin v + \sin v \cos v} \) after simplifying.
• For the right side \( \frac{\tan v - \sin v}{\tan v \sin v} \), logical adjustments and cancellations bring it to \( \frac{1- \cos v}{\sin v} \).

Simplification relies on recognizing patterns and using identities such as the Pythagorean identity or quotient identity as mentioned. Simplified expressions are easier to compare, which aids in verifying the original identity.

Algebraic Manipulation

Algebraic manipulation is the process of rearranging and simplifying expressions. It plays a vital role when verifying trigonometric identities. Manipulating algebraic fractions requires common techniques such as factoring, expanding, and cancelling.

• Apply multiplication, division, or addition/subtraction appropriately to simplify each side of the equation.
• Ensure to multiply the numerator and the denominator by say, \(\cos v\), to avoid fractions within fractions as shown in the left side simplification.
• Cancel common terms wherever possible, like \( \sin v \) in the right side simplification.

By applying these techniques, you maintain the equality of expressions while moving towards a simpler form. Being systematic in your approach to algebraic manipulation unveils the beauty and logic behind the trigonometric identities, allowing for a clearer proof of equivalence.