Verify that the equation is an identity.
$\frac{\sin (x-y)}{\sin (x+y)}=\frac{\tan x-\tan y}{\tan x+\tan y}$
Rewrite the expressions $\sin (x-y)$ and $\sin (x+y)$ using the sine of a difference and sine of a sum identities.
$\frac{\sin (x-y)}{\sin (x+y)}=\frac{\square}{\square}$
Answer (1)
Apply sine subtraction and addition formulas to rewrite the left-hand side: s i n ( x + y ) s i n ( x − y ) = s i n x c o s y + c o s x s i n y s i n x c o s y − c o s x s i n y .
Divide both the numerator and the denominator by cos x cos y to get c o s x s i n x + c o s y s i n y c o s x s i n x − c o s y s i n y .
Simplify the expression using the definition of tangent to obtain t a n x + t a n y t a n x − t a n y .
The identity is verified: s i n ( x + y ) s i n ( x − y ) = t a n x + t a n y t a n x − t a n y .
Explanation
Problem Analysis We are asked to verify the trigonometric identity sin ( x + y ) sin ( x − y ) = tan x + tan y tan x − tan y and to rewrite the left-hand side using the sine of a difference and sine of a sum identities.
Applying Sine Identities We will 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 left-hand side of the identity, we get: sin ( x + y ) sin ( x − y ) = sin x cos y + cos x sin y sin x cos y − cos x sin y
Converting to Tangent To express the fraction in terms of tan x and tan y , we divide both the numerator and the denominator by cos x cos y :
sin x cos y + cos x sin y sin x cos y − cos x sin y = c o s x c o s y s i n x c o s y + c o s x c o s y c o s x s i n y c o s x c o s y s i n x c o s y − c o s x c o s y c o s x s i n y Simplifying, we have: c o s x s i n x + c o s y s i n y c o s x s i n x − c o s y s i n y = tan x + tan y tan x − tan y
Conclusion Thus, we have shown that sin ( x + y ) sin ( x − y ) = tan x + tan y tan x − tan y Therefore, the given equation is an identity.
Examples
In physics, when analyzing wave interference phenomena, this identity can be useful. For example, when two waves with slightly different frequencies interfere, the resulting amplitude can be expressed using sine and cosine functions. This identity helps simplify the expression and relate it to the tangents of the angles, providing a clearer understanding of the interference pattern.
