Using trigonometric formulas, find the value of \(\rm \left(\frac{\sin (x-y)}{\sin(x+y)}\right)\rm \left(\frac{\tan x+\tan y}{\tan x-\tan y}\right)\)
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Understanding the Trigonometric Expression
The problem asks us to find the value of a given trigonometric expression: \( \left(\frac{\sin (x-y)}{\sin(x+y)}\right) \left(\frac{\tan x+\tan y}{\tan x-\tan y}\right) \). To solve this, we need to use standard trigonometric formulas and simplify the expression step by step.
Applying Trigonometric Formulas
We will simplify the two parts of the expression separately and then multiply them.
Part 1: Simplifying \( \frac{\sin (x-y)}{\sin(x+y)} \)
We use the sine subtraction and addition formulas:
- \( \sin(x-y) = \sin x \cos y - \cos x \sin y \)
- \( \sin(x+y) = \sin x \cos y + \cos x \sin y \)
Substituting these into the first part of the expression:
\[ \frac{\sin (x-y)}{\sin(x+y)} = \frac{\sin x \cos y - \cos x \sin y}{\sin x \cos y + \cos x \sin y} \]We can divide both the numerator and the denominator by \( \cos x \cos y \) (assuming \( \cos x \neq 0 \) and \( \cos y \neq 0 \)) to potentially relate it to tangents, but let's keep it in this form for now and look at the second part.
Part 2: Simplifying \( \frac{\tan x+\tan y}{\tan x-\tan y} \)
We use the definition of the tangent function in terms of sine and cosine: \( \tan \theta = \frac{\sin \theta}{\cos \theta} \).
- \( \tan x = \frac{\sin x}{\cos x} \)
- \( \tan y = \frac{\sin y}{\cos y} \)
Substitute these into the second part of the expression:
\[ \frac{\tan x+\tan y}{\tan x-\tan y} = \frac{\frac{\sin x}{\cos x} + \frac{\sin y}{\cos y}}{\frac{\sin x}{\cos x} - \frac{\sin y}{\cos y}} \]To simplify this complex fraction, find a common denominator for the numerator and the denominator, which is \( \cos x \cos y \):
Numerator: \( \frac{\sin x \cos y + \cos x \sin y}{\cos x \cos y} \)
Denominator: \( \frac{\sin x \cos y - \cos x \sin y}{\cos x \cos y} \)
Now, divide the numerator by the denominator:
\[ \frac{\frac{\sin x \cos y + \cos x \sin y}{\cos x \cos y}}{\frac{\sin x \cos y - \cos x \sin y}{\cos x \cos y}} = \frac{\sin x \cos y + \cos x \sin y}{\sin x \cos y - \cos x \sin y} \]Notice that the terms \( \cos x \cos y \) cancel out.
Multiplying the Simplified Parts
Now we multiply the simplified forms of Part 1 and Part 2:
Expression \( = \left(\frac{\sin x \cos y - \cos x \sin y}{\sin x \cos y + \cos x \sin y}\right) \times \left(\frac{\sin x \cos y + \cos x \sin y}{\sin x \cos y - \cos x \sin y}\right) \)
Let \( A = \sin x \cos y - \cos x \sin y \) and \( B = \sin x \cos y + \cos x \sin y \). The expression becomes:
Expression \( = \left(\frac{A}{B}\right) \times \left(\frac{B}{A}\right) \)
Assuming \( A \neq 0 \) and \( B \neq 0 \), we can cancel out the terms:
\[ \frac{A}{B} \times \frac{B}{A} = \frac{A \times B}{B \times A} = 1 \]Thus, the value of the given trigonometric expression is 1.
Conclusion
Using trigonometric formulas for sine difference/sum and the definition of tangent, we simplified the given expression and found its value.
| Original Expression Part | Formula Applied | Simplified Form |
|---|---|---|
| \( \frac{\sin (x-y)}{\sin(x+y)} \) | \( \sin(A \pm B) \) | \( \frac{\sin x \cos y - \cos x \sin y}{\sin x \cos y + \cos x \sin y} \) |
| \( \frac{\tan x+\tan y}{\tan x-\tan y} \) | \( \tan \theta = \frac{\sin \theta}{\cos \theta} \) | \( \frac{\sin x \cos y + \cos x \sin y}{\sin x \cos y - \cos x \sin y} \) |
| Product of parts | Multiplication & Cancellation | \( 1 \) |
Revision Table: Key Trigonometric Identities
| Identity | Formula |
|---|---|
| Sine Sum Formula | \( \sin(A+B) = \sin A \cos B + \cos A \sin B \) |
| Sine Difference Formula | \( \sin(A-B) = \sin A \cos B - \cos A \sin B \) |
| Tangent Identity | \( \tan \theta = \frac{\sin \theta}{\cos \theta} \) |
Additional Information: Understanding Trigonometric Identities
Trigonometric identities are equations that are true for all values of the variables for which the expressions are defined. They are fundamental tools in trigonometry for simplifying expressions, solving equations, and proving other identities.
The sum and difference formulas for sine and cosine are derived geometrically or using Euler's formula and are essential for expanding or simplifying expressions involving sums or differences of angles. The tangent identity is a ratio identity, directly following from the definitions of sine, cosine, and tangent in a right-angled triangle or on the unit circle.
Using these identities correctly allows us to transform complex trigonometric expressions into simpler forms, as demonstrated in this problem, where the entire expression simplifies to a constant value of 1.
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