Question

eliminate theta if tan( theta -alpha) =a and tan (theta alpha)=b​

Answer 1

Answer:

Tan2α  = a + b

Step-by-step explanation:

eliminate theta if tan( theta -alpha) =a and tan (theta alpha)=b​

Tan(θ - α)  = a

Tan(θ + α) = b

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

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

Tan(θ - α)  =   (Tanθ - Tanα)/(1 + TanθTanα)  = a

=> Tanθ - Tanα = a  + aTanθTanα

=> Tanθ(1 + aTanα) = a + Tanα

=> Tanθ =  (a + Tanα)/(1 + aTanα)

Tan(θ + α) =  (Tanθ + Tanα)/(1 - TanθTanα) = b

=> Tanθ + Tanα = b  - bTanθTanα

=> Tanθ(1 + b Tanα) = b - Tanα

=> Tanθ =  (b - Tanα)/(1 + bTanα)

Equating Tanθ

(a + Tanα)/(1 + aTanα)  =  (b - Tanα)/(1 + bTanα)

=>  (a + Tanα)(1 + bTanα) =  (1 + aTanα) (b - Tanα)

=> a + Tanα + abTanα + bTan²α  =  b - Tanα  + abTanα  - aTan²α

=> 2Tanα  + (a + b)Tan²α =  (b + a)

=> 2 Tanα = (b+a)( 1  - Tan²α)

=> b + a =  2 Tanα/ ( 1  - Tan²α)

=> b + a = (Tan α + Tanα)/(1  - TanαTanα)

=> b + a = Tan(α + α)

=> b + a = Tan(2α)

Tan2α  = a + b

Answer 2

Step-by-step explanation:

Kindly refer to the attachments.

Hope it helps.