Problem 35 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The identity is verified: both sides simplify to \(-\csc(\alpha)\).

Step 1: Use Trigonometric Identities

Start by rewriting the trigonometric functions with negative angles using known identities. \- Using the identity \( \cot(-\alpha) = -\cot(\alpha) \), rewrite \( \cot(-\alpha) \) as \(-\cot(\alpha)\). \- Using the identity \( \cos(-\alpha) = \cos(\alpha) \), rewrite \( \cos(-\alpha) \) as \( \cos(\alpha) \). \- Using the identity \( \sin(-\alpha) = -\sin(\alpha) \), rewrite \( \sin(-\alpha) \) as \(-\sin(\alpha)\). \Thus, the left-hand side becomes: \[-\cot(\alpha) \cdot \cos(\alpha) - \sin(\alpha)\]

Step 2: Simplify the Expression

Next, substitute the identity \( \cot(\alpha) = \frac{\cos(\alpha)}{\sin(\alpha)} \) into the expression. \The expression becomes: \[-\left(\frac{\cos(\alpha)}{\sin(\alpha)}\right) \cos(\alpha) - \sin(\alpha)\]

Step 3: Combine and Simplify Terms

Now, combine the terms on the left-hand side of the equation: \- Multiply \(-\left(\frac{\cos(\alpha)}{\sin(\alpha)}\right) \cos(\alpha)\) to get \( -\frac{\cos^2(\alpha)}{\sin(\alpha)} \). \- Combine this with \(-\sin(\alpha)\) over a common denominator, which is \( \sin(\alpha) \): \\[-\frac{\cos^2(\alpha)}{\sin(\alpha)} - \frac{\sin^2(\alpha)}{\sin(\alpha)} = \frac{-(\cos^2(\alpha) + \sin^2(\alpha))}{\sin(\alpha)}\] \- Use the Pythagorean identity \( \cos^2(\alpha) + \sin^2(\alpha) = 1 \) to simplify to \(-\frac{1}{\sin(\alpha)}\). \Thus, the expression simplifies to \(-\csc(\alpha)\).

Step 4: Compare with Right-Hand Side

Finally, compare the simplified left-hand side with the right-hand side of the original identity. Both sides simplify to \(-\csc(\alpha)\), confirming the identity.

Key Concepts

Cotangent Function

The cotangent function is a fundamental trigonometric function. It is the reciprocal of the tangent function. In mathematical terms, cotangent is defined as the ratio of the adjacent side to the opposite side in a right triangle. The cotangent of an angle \( \alpha \) is expressed as:

• \( \cot(\alpha) = \frac{1}{\tan(\alpha)} \)
• Alternatively, it can also be expressed using sine and cosine as \( \cot(\alpha) = \frac{\cos(\alpha)}{\sin(\alpha)} \)

This means that the cotangent function relates the cosine and sine of an angle.
In our context, applying this understanding helps simplify trigonometric expressions by breaking them into more basic components. This simplification is crucial when proving identities, like transforming \(-\cot(\alpha) \cdot \cos(\alpha)\) in our verification task.

Cosecant Function

The cosecant function, abbreviated as csc, is another reciprocal trigonometric function. Specifically, it is the reciprocal of the sine function. For an angle \( \alpha \), the cosecant is defined as:

• \( \csc(\alpha) = \frac{1}{\sin(\alpha)} \)

This relationship indicates that wherever the sine is zero, the cosecant is undefined, since division by zero is undefined in mathematics.
In the given exercise where we need to show that an expression equals \(-\csc(\alpha)\), understanding the nature of the cosecant function helps clarify the type of output expected from the calculation. The equivalent expression of \(-\csc(\alpha)\) highlights the role of this reciprocal function in particular angles.

Negative Angle Identities

Negative angle identities are particular trigonometric rules that relate to the sine, cosine, and tangent functions with negative angles. These identities allow angles to be expressed positively in calculations, simplifying the analysis of trigonometric values at negative angles:

• \( \sin(-\alpha) = -\sin(\alpha) \)
• \( \cos(-\alpha) = \cos(\alpha) \)
• \( \tan(-\alpha) = -\tan(\alpha) \)

Using these identities is essential for verifying trigonometric identities, as they allow for a standardized approach to handling negative angles while solving problems.
In the problem at hand, we used these identities to rewrite \( \cot(-\alpha), \cos(-\alpha), \) and \( \sin(-\alpha) \) into more manageable terms, aiding in the simplification and verification of the given trigonometric identity.