Problem 52 - Algebra 2

Question

Verify that each of the following is an identity. $$ \cos \theta(\cos \theta+\cot \theta)=\cot \theta \cos \theta(\sin \theta+1) $$

Step-by-Step Solution

Answer

The expression is an identity as both sides are equal.

Step 1: Simplify the Left-Hand Side (LHS)

We start with the left-hand side: \( \cos \theta (\cos \theta + \cot \theta) \).Substitute \( \cot \theta = \frac{\cos \theta}{\sin \theta} \),LHS becomes:\[ \cos \theta \left( \cos \theta + \frac{\cos \theta}{\sin \theta} \right) = \cos \theta \left( \frac{\cos \theta \sin \theta + \cos \theta}{\sin \theta} \right) \]\[ = \cos \theta \frac{\cos \theta (1 + \sin \theta)}{\sin \theta} \] This simplifies to:\[ = \frac{\cos^2 \theta (1 + \sin \theta)}{\sin \theta} \].

Step 2: Simplify the Right-Hand Side (RHS)

Now consider the right-hand side: \( \cot \theta \cos \theta(\sin \theta + 1) \).Substituting \( \cot \theta = \frac{\cos \theta}{\sin \theta} \),RHS becomes:\[ \frac{\cos \theta}{\sin \theta} \cos \theta (\sin \theta + 1) \]\[ = \frac{\cos^2 \theta (\sin \theta + 1)}{\sin \theta} \].

Step 3: Compare LHS and RHS

Both sides of the equation are now expressed as:\[ \frac{\cos^2 \theta (1 + \sin \theta)}{\sin \theta} \].Thus, the left-hand side is equal to the right-hand side, verifying that the expression is indeed an identity.

Key Concepts

Cosine Function

The cosine function is a fundamental component in trigonometry, often represented as \( \cos \theta \). It calculates the adjacent side of a right triangle divided by its hypotenuse. Understanding the properties of the cosine function is essential in many mathematical calculations, especially when dealing with trigonometric identities.

• **Basic Definition**: It is the ratio of the length of the adjacent side to the length of the hypotenuse in a right triangle.
• **Range and Period**: The function oscillates between -1 and 1, making it periodic with a period of \( 2\pi \).
• **Common Values**: Knowing values like \( \cos 0 = 1 \), \( \cos \frac{\pi}{2} = 0 \), and \( \cos \pi = -1 \) helps solve complex trigonometric problems.

In the original exercise, the cosine function is repeatedly used to simplify both sides of the equation. Its contribution is crucial as it allows the algebraic manipulation which leads to the simplification of the identity.

Cotangent Function

The cotangent function, expressed as \( \cot \theta \), is another key player in trigonometry. It is derived from the tangent function and is the reciprocal of the tangent. This particular function is defined as the ratio of the adjacent side to the opposite side in a right triangle.

• **Relationship with Other Functions**: Unlike tangent, which is opposite over adjacent, cotangent is adjacent over opposite or, more simply, \( \cot \theta = \frac{\cos \theta}{\sin \theta} \).
• **Properties**: Its values are undefined at multiples of \( \pi \), making it important to consider its domain when working on problems.
• **Usage in Identities**: As seen in the identity verification, substituting \( \cot \theta \) with \( \frac{\cos \theta}{\sin \theta} \) simplifies the trigonometric equations significantly.

By substituting cotangent in the exercise, it helped to transform the problem into a simpler form, leading to verification of the identity.

Sine Function

The sine function, denoted as \( \sin \theta \), is one of the primary trigonometric functions alongside cosine and tangent. It measures the ratio of the opposite side to the hypotenuse in a right triangle. Its role is significant in equations, especially when identities are involved.

• **Basic Definition**: The sine of an angle is the ratio of the length of the side of the triangle opposite the angle to the length of the triangle's hypotenuse.
• **Range and Period**: Like the cosine function, sine values range from -1 to 1 and have a periodicity of \( 2\pi \).
• **Complementary Relation**: Sine and cosine are complementary, meaning for any angle \( \theta \), \( \sin(\frac{\pi}{2} - \theta) = \cos \theta \).

In our specific trigonometric identity, \( \sin \theta \) appears in both the left and right sides of the equation, allowing for common factors and ultimately, the successful validation of the identity.