Problem 22 - Algebra 2

Question

Verify that each of the following is an identity. $$ \sin 2 x=2 \cot x \sin ^{2} x $$

Step-by-Step Solution

Answer

The identity \( \sin 2x = 2 \cot x \sin^2 x \) is true.

Step 1: Understand the Identity to Verify

We need to verify the identity \( \sin 2x = 2 \cot x \sin^2 x \). This means we need to show that the left-hand side (LHS) is equal to the right-hand side (RHS) for all values of \( x \) where both sides are defined.

Step 2: Recall and Substitute Double Angle Identity

Recall that \( \sin 2x = 2 \sin x \cos x \). Substitute this into the LHS: \( \sin 2x = 2 \sin x \cos x \).

Step 3: Rewrite \( \cot x \) in Terms of \( \sin x \) and \( \cos x \)

Remember that \( \cot x = \frac{\cos x}{\sin x} \). Substitute this into the RHS: \( 2 \cot x \sin^2 x = 2 \left( \frac{\cos x}{\sin x} \right) \sin^2 x \).

Step 4: Simplify the RHS Expression

Simplify the RHS: \[ 2 \left( \frac{\cos x}{\sin x} \right) \sin^2 x = 2 \cos x \sin x \].

Step 5: Compare and Conclude

After simplifying, the RHS is \( 2 \cos x \sin x \) which matches the LHS \( 2 \sin x \cos x \). Hence, the identity is verified as both sides are equal.

Key Concepts

Double Angle Identity

The double angle identity for sine is essential when analyzing trigonometric equations like \( \sin 2x = 2 \cot x \sin^2 x \). This identity, \( \sin 2x = 2 \sin x \cos x \), helps us to transform equations by doubling the angle, a technique often used to simplify trigonometric identities. It allows us to relate a single trigonometric function of a double angle to the product of sine and cosine.

When working with equations involving trigonometric identities, recognizing and correctly applying the double angle identity is crucial to simplifying and solving complex expressions. It is a fundamental part of trigonometry often involving problems where combining or expanding angles precisely leads to simplified results.

The double angle identity provides a direct method to verify or transform expressions without necessarily solving them fully. It's one of several identities used to reframe and simplify trigonometric relationships, making it a valuable tool in mathematics.

Cotangent

In trigonometry, the cotangent function is the reciprocal of the tangent function. Expressed mathematically, it is \( \cot x = \frac{1}{\tan x} = \frac{\cos x}{\sin x} \).

Cotangent is crucial when dealing with equations where dividing sine and cosine comes into play. By converting cotangent to the ratio of cosine to sine, as seen in \( 2 \cot x \sin^2 x \), we can simplify expressions significantly.

Understanding the relationships between sine, cosine, and cotangent is valuable for simplifying trigonometric identities. This concept is used to transform equations and verify expressions by expressing one trigonometric function in terms of others, which can often lead to easier simplification.

When solving problems, rewriting cotangent in terms of sine and cosine is a common technique that provides a clearer pathway for manipulation and simplification.

Sine

The sine function, denoted as \( \sin x \), is a fundamental trigonometric function representing the ratio of the opposite side of a right triangle to its hypotenuse.

Sine is not just a simple ratio; it is integral in analyzing and deriving numerous trigonometric identities, such as the often-used double angle identity \( \sin 2x = 2 \sin x \cos x \).

In the context of the original expression \( \sin 2x = 2 \cot x \sin^2 x \), sine plays a vital role in simplifying the equations. Understanding that sine is involved in both the left-hand and right-hand sides of the equation is critical for verification or simplification purposes.

Recognizing how sine functions within the context of an equation allows us to substitute, simplify, and validate identities more effectively. This familiarity with sine and its transformations aids in developing a more comprehensive understanding of trigonometric identities.

Cosine

The cosine function, represented as \( \cos x \), is another primary trigonometric function. It is defined as the ratio of the adjacent side to the hypotenuse in a right triangle.

Cosine is often paired with sine in many trigonometric identities and plays a crucial role in expressions like the double angle identity \( \sin 2x = 2 \sin x \cos x \). In the equation verification, cosine is necessary for expressing and simplifying identities successfully.

When dealing with identities such as \( \sin 2x = 2 \cot x \sin^2 x \), understanding the role of cosine makes simplification possible, especially considering transformations, like double angles, or conversions between other trigonometric functions.

Recognizing cosine’s role in these relationships allows for ease in transitioning between expressions and understanding complex identities, further supporting verification endeavors.