Question

If 2 sin x = sin y and 2 cos x = 3 cos y then find the value of tan (x + y)

Answer 1

Answer:

Tan(x + y) = √15

Step-by-step explanation:

If 2 sin x = sin y and 2 cos x = 3 cos y then find the value of tan (x + y)

Dividing given equations :

2Sinx/2Cosx = Siny/3Cosy

=>  Tanx = Tany/3

=> Tany = 3 Tanx

Squaring given Equation

4Sin²x = Sin²y

4 Cos²x = 9Cos²y

Adding both

4 = Sin²y + 9 Cos²y

=> 4 = 1 + 8Cos²y

=> 3 = 8 Cos²y

=> Sec²y = 8/3

=> 1 + Tan²Y = 8/3

=> Tan²y = 5/3

=> Tany = √5/√3

Tan(x + y)  = (Tan x + Tany )/( 1 - TanxTany)

= (Tany/3 + Tany)(1 - Tan²Y/3)

= 4 Tany /(3 - Tan²y)

= 4 Tany /( 3 - 5/3)

= 12 Tany (4)

= 3 Tan y

= 3  √5 / √3

= √3 * √5

= √15