Question

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

Accepted Answer

Start by recalling the definitions of cotangent in terms of sine and cosine: $$\cot x = \frac{\cos x}{\sin x}$$ and $$\cot y = \frac{\cos y}{\sin y}. $$Rewrite the right-hand side (RHS) of the equation $$\frac{\cot x + \cot y}{1 + \cot x \cot y} by $$substituting the cotangent expressions: $$\frac{\frac{\cos x}{\sin x} + \frac{\cos y}{\sin y}}{1 + \frac{\cos x}{\sin x} \cdot \frac{\cos y}{\sin y}}. $$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 $$\frac{\sin x \sin y + \cos x \cos y}{\sin x \sin y}. $$Simplify the complex fraction by multiplying numerator and denominator by $$\sin x \sin y$$, which will cancel the denominators inside the fraction, resulting in $$\frac{\cos x \sin y + \cos y \sin x}{\sin x \sin y + \cos x \cos y}. $$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.

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

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

Trigonometric Identities

Angle Sum and Difference Formulas

Cotangent and Its Relationship to Sine and Cosine