Question

Prove : 1+tan theta/ 1- tan theta + 1- tan theta / 1+ tan theta = 2sec²theta /2- sec² theta​

Answer 1

Answer:

Step-by-step explanation:

1+tanθ/ 1- tanθ + 1- tanθ / 1+ tanθ

=> 1 + Sinθ/Cosθ / 1 - Sinθ/Cosθ + 1 - Sinθ/Cosθ/1 + Sinθ/Cosθ

=> Cosθ + Sinθ/Cosθ/Cosθ-Sinθ/Cosθ + Cosθ-Sinθ/Cosθ/Cosθ + Sinθ/Cosθ

=> Cosθ + Sinθ/Cosθ-Sinθ + Cosθ-Sinθ/Cosθ + Sinθ

=> (Cosθ + Sinθ)² + (Cosθ-Sinθ)²/(Cosθ-Sinθ)(Cosθ + Sinθ)

=> 2/Cos²θ - Sin²θ

=> 2/Cos²θ - 1 + Cos²θ  (∵ Sin²θ = 1 - Cos²θ)

=> 2/2Cos²θ - 1

//Multiply numerator and denominator by Sec²θ

=> 2Sec²θ/(2Cos²θ - 1)Sec²θ

=> 2Sec²θ/ 2 - Sec²θ   (∵ Cos²θ * Sec²θ = 1)

=> R.H.S

Hence proved