Sum

Prove the following identities, where the angles involved are acute angles for which the expressions are defined.

`((1+tan^2A)/(1+cot^2A))=((1-tanA)/(1-cotA))^2=tan^2A`

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#### Solution

`((1+tan^2A)/(1+cot^2A))=((1-tanA)/(1-cotA))^2=tan^2A`

`(1+tan^2A)/(1+cot^2A)=(1+sin^2A/cos^2A)/(1+cos^2A/sin^2A)=((cos^2A+sin^2A)/cos^2A)/((sin^2A+cos^2A)/sin^2A)`

`=(1/cos^2A)/(1/sin^2A)=sin^2A/cos^2A`

`=tan^2A`

`((1-tanA)/(1-cotA))^2=(1+tan^2A-2tanA)/(1+cot^2A-2cotA)`

`=(sec^2A-2tanA)/(cosec^2A-2cotA)`

`=(1/cos^2A-(2sinA)/cosA)/(1/sin^2A-(2cosA)/sinA)=((1-2sinAcosA)/cos^2A)/((1-2sinAcosA)/sin^2A)`

`=sin^2A/cos^2A=tan^2A`

Concept: Trigonometry

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