Using trigonometric formulas, find the value of \(\rm \left(\frac{\sin (x-y)}{\sin(x+y)}\right)\rm \left(\frac{\tan x+\tan y}{\tan x-\tan y}\right)\)

Formula used:

Sin (A - B) = sin A × cos B - cos A × sin B

Sin (A + B) = sin A × cos B + cos A × sin B

Calculation:

\(\rm \left(\frac{\sin (x-y)}{\sin(x+y)}\right)\rm \left(\frac{\tan x+\tan y}{\tan x-\tan y}\right)\)

⇒ (sin x × cos y - cos x × sin y)/(sin x × cos y + cos x × sin y) × {(sin x/cos x) + (sin y/cos y)}/{(sin x/cos x) + (sin y/cos y)}

⇒ (sin x × cos y - cos x × sin y)/(sin x × cos y + cos x × sin y) × {(sin x × cos y) + (sin y × cos x/(cos x) × (cos y)}/{(sinx × cos y) + (sin y × cos x)/(cos x) × (cosy)}

⇒ (sin x × cos y - cos x × sin y)/(sin x × cos y + cos x × sin y) × (sin x × cos y) + (siny × cos x)/(sin x × cos y) + (sin y × cos x)

⇒ 1

∴ The correct answer is 1.

Alternate MethodCalculation:

if we put x = 45° and y = 0° then,

\(\rm \left(\frac{\sin (x-y)}{\sin(x+y)}\right)\rm \left(\frac{\tan x+\tan y}{\tan x-\tan y}\right)\) 

⇒ sin (45 - 0)/sin(45 + 0) × {(tan 45 + tan 0)/(tan 45 - tan 0)}

⇒ sin 45/sin 45 × {(1 + 0)/(1 - 0)} 

⇒ 1

∴ The correct answer is 1.