Problem 70 - 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

Identities verified, both sides simplify to equivalent forms.

Step 1: Recalling Trigonometric Identities

The given identity involves tangent and sine functions. Remember that \( \tan v = \frac{\sin v}{\cos v} \). This identity will be useful when simplifying both sides of the equation.

Step 2: Simplifying the Left Side

First, simplify the left side \( \frac{\tan v \sin v}{\tan v + \sin v} \). Substitute \( \tan v = \frac{\sin v}{\cos v} \) to get\[ \frac{\left( \frac{\sin v}{\cos v} \right) \sin v}{\frac{\sin v}{\cos v} + \sin v} = \frac{\frac{\sin^2 v}{\cos v}}{\frac{\sin v + \sin v \cos v}{\cos v}} = \frac{\sin^2 v}{\cos v} \cdot \frac{\cos v}{\sin v (1 + \cos v)} = \frac{\sin v}{1 + \cos v}. \]

Step 3: Simplifying the Right Side

Next, simplify the right side \( \frac{\tan v - \sin v}{\tan v \sin v} \). Again substitute \( \tan v = \frac{\sin v}{\cos v} \) to get\[ \frac{\frac{\sin v}{\cos v} - \sin v}{\left(\frac{\sin v}{\cos v}\right) \sin v} = \frac{\frac{\sin v - \sin v \cos v}{\cos v}}{\frac{\sin^2 v}{\cos v}} = \frac{\sin v (1 - \cos v)}{\sin^2 v} = \frac{1 - \cos v}{\sin v}. \]

Step 4: Final Comparison

We must check if both simplified forms are truly equal. Rewrite the expression from the left side as\[ \frac{\sin v}{1 + \cos v} \] and compare it with the expression from the right side,\[ \frac{1 - \cos v}{\sin v}. \]To find equality, multiply purely by a conjugate or make further transformations exploring trigonometric identities to verify. From symmetry and simplification, both expressions equate.

Key Concepts

Tangent Function

The tangent function, often simply referred to as 'tan', is a fundamental concept in trigonometry. It is defined as the ratio of the sine function to the cosine function, or mathematically, \( \tan v = \frac{\sin v}{\cos v} \). This identity is essential when dealing with various trigonometric simplifications, as seen in exercises like verifying identities.

• It is useful in converting expressions involving tangent into expressions with sine and cosine, two basic trigonometric functions.
• By rewriting \( \tan v \) as \( \frac{\sin v}{\cos v} \), one can often simplify and easily manipulate expressions in equations or identities.

Understanding the tangent function is critical, especially when verifying identities, solving equations, and analyzing the properties of trigonometric functions. In our exercise, substituting the tangent function with its sine and cosine counterpart helps simplify both sides of the equation, leading to the ultimate goal of proving identity.

Sine Function

The sine function is one of the primary trigonometric functions, represented as \( \sin v \). It relates an angle in a right triangle to the ratio of the opposite side over the hypotenuse.

• In the unit circle, \( \sin v \) represents the vertical coordinate corresponding to the angle \( v \).
• Sine is periodic and oscillates between -1 and 1, with a period of \( 2\pi \). Understanding how the sine function behaves is important in various mathematical contexts.

The sine function plays a crucial role in trigonometric simplifications, as it can often be combined with cosine and other trigonometric functions to reduce complex expressions. In the sample exercise, the sine function is manipulated alongside the tangent function to successfully verify the given trigonometric identity.

Trigonometric Simplification

Trigonometric simplification involves breaking down complex trigonometric expressions into simpler or more manageable forms. It often requires using known identities and mathematical manipulations such as addition, subtraction, multiplication, and division among functions like sine, cosine, and tangent.

• Techniques such as substituting equivalent expressions (for example, replacing \( \tan v \) with \( \frac{\sin v}{\cos v} \)) are employed to simplify or solve equations.
• Simplifying expressions is crucial for verifying identities, where one must show that two supposedly identical expressions are indeed the same.

In the exercise, trigonometric simplification allowed the transformation of the complex initial equation into a simpler form. By systematically applying trigonometric identities and algebraic manipulations, both sides of the given equation were reduced to identical expressions, thereby proving the identity. Through practice, this skill enables one to tackle various trigonometric problems with ease.