Problem 44 - Contemporary Precalculus

Question

$$\text { Prove the identity.}$$ $$\frac{\sin (x-y)}{\sin x \cos y}=1-\cot x \tan y$$

Step-by-Step Solution

Answer

Question: Prove the trigonometric identity: $\frac{\sin(x-y)}{\sin x \cos y} = 1 - \cot x \tan y$. Solution: Follow the steps in the given solution, and rewrite the left side of the equation using trigonometric identities, simplify the expression, and compare with the right side. If both sides are equal, then the identity is proved. In this case, the identity is indeed true: $$\frac{\sin(x-y)}{\sin x \cos y} = 1 - \cot x \tan y$$

Step 1: Rewrite Using Trigonometric Identities

Since we have a sine function in the numerator and two trigonometric functions in the denominator, we may want to consider using sine and cosine relationships. Let's rewrite the left side of the equation using the difference of angles identities: $$\frac{\sin(x-y)}{\sin x \cos y} = \frac{\sin x \cos y - \cos x \sin y}{\sin x \cos y}$$

Step 2: Simplify the Left Side

We can simplify the left side by canceling out some terms: $$\frac{\cancel{\sin x \cos y} - \cos x \sin y}{\cancel{\sin x \cos y}} = 1 - \frac{\cos x \sin y}{\sin x \cos y}$$

Step 3: Rewrite the Remaining Fraction Using Trigonometric Identities

Again, let's use the definitions of cotangent and tangent to rewrite the remaining fraction: $$1 - \frac{\cos x \sin y}{\sin x \cos y} = 1 - \frac{\frac{\cos x}{\sin x}}{\frac{\sin y}{\cos y}}$$ Now, we can rewrite the fraction once more using the cotangent definition. $$1 - \frac{\cot x}{\tan y} = 1 - \cot x \tan y$$ So, we have proved the identity: $$\frac{\sin (x-y)}{\sin x \cos y} = 1 - \cot x \tan y$$

Key Concepts

Sine and Cosine Relationships

Understanding sine and cosine relationships is fundamental to grasping many trigonometric identities. Essentially, sine and cosine are functions of an angle and serve as the foundation of trigonometry.
- **Sine** (\(\sin(\theta)\)) is the ratio of the opposite side to the hypotenuse in a right triangle.- **Cosine** (\(\cos(\theta)\)) is the ratio of the adjacent side to the hypotenuse.
These relationships are not only used to find the values of angles and sides but also to simplify expressions. In this exercise, we used them to rewrite the sine of an angle difference.
The sine of a difference, \(\sin(x-y)\), can be expanded using the identity:\[\sin(x-y) = \sin x \cos y - \cos x \sin y\]
This allows us to replace \(\sin(x-y)\) in the given expression, setting the stage for further simplifications.

Cotangent and Tangent

The cotangent and tangent functions are closely related to sine and cosine, providing alternative ways to express trigonometric ratios.
- **Tangent** (\(\tan(\theta)\)) is defined as \(\frac{\sin(\theta)}{\cos(\theta)}\).- **Cotangent** (\(\cot(\theta)\)) is the reciprocal, defined as \(\frac{1}{\tan(\theta)}\) or more simply as \(\frac{\cos(\theta)}{\sin(\theta)}\).
These identities are invaluable when simplifying expressions. In our problem, after expanding and simplifying \(\sin(x-y)\), we rewrote the remaining terms using cotangent and tangent identities to further streamline the equation. This process showed the equivalency between the expanded form and the target identity expression.

Angle Difference Identities

Angle difference identities help simplify the expression involving trigonometric functions of angle differences.
Using such identities, complex interrelationships between these angles can be broken down or expanded. For sine, the angle difference identity is:\[\sin(x-y) = \sin x \cos y - \cos x \sin y\]For cosine, it's given by:\[\cos(x-y) = \cos x \cos y + \sin x \sin y\]
In this exercise, we used the sine difference identity to express \(\sin(x-y)\) as a combination of products of sine and cosine. This allowed us to manipulate the original expression smoothly, ultimately reaching the simplified form using cotangent and tangent.Understanding these identities is crucial when dealing with proofs or simplifications, as they provide convenient shortcuts to more straightforward expressions involved in solving trigonometric equations.