Verify that each equation is an identity.
sin(x + y)/cos(x - y) = (cot x + cot y)/(1 + cot x cot y)

Verified step by step guidance

1. Start by recalling the definitions of cotangent in terms of sine and cosine: \(\cot x = \frac{\cos x}{\sin x}\) and \(\cot y = \frac{\cos y}{\sin y}\).
2. Rewrite the right-hand side (RHS) of the equation \(\frac{\cot x + \cot y}{1 + \cot x \cot y}\) by substituting the cotangent expressions: \(\frac{\frac{\cos x}{\sin x} + \frac{\cos y}{\sin y}}{1 + \frac{\cos x}{\sin x} \cdot \frac{\cos y}{\sin y}}\).
3. Find a common denominator for the numerator and denominator of the RHS to combine the fractions: numerator becomes \(\frac{\cos x \sin y + \cos y \sin x}{\sin x \sin y}\) and denominator becomes \(\frac{\sin x \sin y + \cos x \cos y}{\sin x \sin y}\).
4. Simplify the complex fraction by multiplying numerator and denominator by \(\sin x \sin y\), which will cancel the denominators inside the fraction, resulting in \(\frac{\cos x \sin y + \cos y \sin x}{\sin x \sin y + \cos x \cos y}\).
5. Recognize the trigonometric sum formulas: numerator \(\cos x \sin y + \cos y \sin x = \sin(x + y)\) and denominator \(\sin x \sin y + \cos x \cos y = \cos(x - y)\). Thus, the RHS simplifies to \(\frac{\sin(x + y)}{\cos(x - y)}\), which matches the left-hand side (LHS), verifying the identity.

Key Concepts

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values within their domains. Verifying an identity means showing both sides simplify to the same expression using known formulas, such as angle sum and difference identities.

Angle Sum and Difference Formulas

These formulas express trigonometric functions of sums or differences of angles in terms of functions of individual angles. For example, sin(x + y) = sin x cos y + cos x sin y and cos(x - y) = cos x cos y + sin x sin y, which are essential for rewriting and simplifying expressions.

Cotangent and Its Relationship to Sine and Cosine

Cotangent is the reciprocal of tangent, defined as cot θ = cos θ / sin θ. Understanding how to rewrite cotangent in terms of sine and cosine helps in transforming and simplifying expressions involving cot x and cot y to verify identities.