Question

Prove the following:

`((1 - tan θ)/(1 - cot θ))^2 = tan^2θ`

Answer 1

⇒ `((1 - tan θ)/(1 - cot θ))^2`

⇒ `((1 - tan θ)/(1/1 - 1/tan θ))^2`

⇒ `((1 - tan θ)/(((tan θ - 1)/tan θ)))^2`

⇒ `((1 - tan θ) xx tan θ/(tan θ - 1))^2`

⇒ `((1 - tan θ)^2 tan^2 θ)/(tan - 1)^2`

(a − b)² = a² − 2ab + b²

(b − a)² = b² − 2ab + a²

⇒ tan² θ

`((1 - tan θ)/(1 - cot θ))^2 = ((1/((1)) - sin θ/((cos θ)))/(1/((1)) - cos θ/((sin θ))))^2`

= `(((cos θ - sin θ)/cos θ)/((sin θ - cos θ)/sin θ))^2`

= `((cos θ - sin θ)/cos θ xx sin θ/(sin θ - cos θ))^2`

= `((cos θ - sin θ)^2 xx sin^2 θ)/(cos^2 θ (sin θ - cos θ)^2)`

= `sin^2 θ/cos^2 θ`

= tan² θ