Problem 45 - Algebra and Trigonometry

Question

Verify the identity. $$ (\cot x-\csc x)(\cos x+1)=-\sin x $$

Step-by-Step Solution

Answer

The identity is verified as both sides equal \(-\sin x\).

Step 1: Recall Trigonometric Identities

Recall that \( \cot x = \frac{\cos x}{\sin x} \) and \( \csc x = \frac{1}{\sin x} \). These will be used to express the left-hand side in terms of sine and cosine.

Step 2: Rewrite in Terms of Sine and Cosine

Substitute the identities from Step 1 into the expression \((\cot x - \csc x)(\cos x + 1)\) to get:\[ \left( \frac{\cos x}{\sin x} - \frac{1}{\sin x} \right)(\cos x + 1) \]This simplifies to:\[ \left( \frac{\cos x - 1}{\sin x} \right)(\cos x + 1) \]

Step 3: Simplify the Product

Expand the expression to simplify:\[ (\cos x - 1)(\cos x + 1) = \cos^2 x - 1 \]Using the identity \( \cos^2 x - 1 = -\sin^2 x \), we can write:\[ \frac{-\sin^2 x}{\sin x} \]

Step 4: Complete the Simplification

Simplify the fraction \( \frac{-\sin^2 x}{\sin x} = -\sin x \). Thus, the original expression becomes:\[ -\sin x \]

Step 5: Verify Identity

Since we simplified the left-hand side to \(-\sin x\), it matches the right-hand side of the equation. Therefore, the identity is verified.

Key Concepts

Cotangent

The cotangent, often denoted as \( \cot x \), is one of the six fundamental trigonometric functions. It is defined as the reciprocal of the tangent function. In terms of the sine and cosine functions, cotangent is expressed as the ratio of \( \cos x \) to \( \sin x \). This can be represented mathematically as:

• \( \cot x = \frac{\cos x}{\sin x} \)

This identity is particularly useful when simplifying trigonometric expressions. By rewriting cotangent in terms of sine and cosine, complex expressions become more manageable. Whether you are simplifying equations or trying to prove identities, remember the cotangent as a ratio of cosine to sine.

Cosecant

Cosecant, notated as \( \csc x \), is another essential trigonometric function. It is defined as the reciprocal of the sine function, which means:

• \( \csc x = \frac{1}{\sin x} \)

Cosecant, like cotangent, is crucial for expressing trigonometric equations in simpler terms.For the exercise solution, expressing \( \csc x \) in terms of sine allowed us to rewrite the initial expression using a common denominator, aiding in the process of simplification. Understanding these relationships is key for solving trigonometric equations efficiently.

Simplifying Expressions

Simplifying trigonometric expressions involves reducing complex trigonometric formulas to their simplest form and can often involve using trigonometric identities. In the provided exercise, we started by expressing trigonometric functions like \( \cot x \) and \( \csc x \) in terms of sine and cosine. This led to:

• \( (\cot x - \csc x)(\cos x + 1) = \left( \frac{\cos x - 1}{\sin x} \right)(\cos x + 1) \)

The next step was to expand and simplify the resulting equation by applying the trigonometric identity:

• \( \cos^2 x - 1 = -\sin^2 x \)

This transformation made it easier to simplify the expression to \( -\sin x \). Always look to utilize identities and substitute wherever possible to break down the steps methodically.

Trigonometric Functions

Trigonometric functions are the building blocks of trigonometry, frequently appearing in various mathematical contexts, from geometry to calculus. These include sine, cosine, tangent, cotangent, secant, and cosecant.Understanding each function's definition is vital. For example:

• Sine: \( \sin x \) is the ratio of the opposite side to the hypotenuse in a right triangle.
• Cosine: \( \cos x \) is the ratio of the adjacent side to the hypotenuse.
• Tangent: \( \tan x = \frac{\sin x}{\cos x} \)
• Cotangent: \( \cot x = \frac{\cos x}{\sin x} \)
• Secant: \( \sec x = \frac{1}{\cos x} \)
• Cosecant: \( \csc x = \frac{1}{\sin x} \)

Each of these functions has a corresponding identity that can help simplify and solve trigonometric problems. Familiarization with these relationships allows for easier manipulation of expressions, as seen in the exercise's solution, which utilized these relationships to achieve the final form.