`(1-tanx )^2 + (1-cotx )^2 `

`= 1-2tanx+tan^2x+1-2cotx+cot^2x`

`= (1+tan^2x)+(1+cot^2x)-2(tanx+cotx)`

We know that;

`1+tan^2x = sec^2x`

`1+cot^2x = cosec^2x`

`sin^2x+cos^2x = 1`

`(tanx+cotx)`

`= sinx/cosx+cosx/sinx`

`= (sin^2x+cos^2x)/(sinxcosx)`

`= 1/(sinxcosx)`

`= secxcosecx`

`(1-tanx )^2 + (1-cotx )^2 `

`= (1+tan^2x)+(1+cot^2x)-2(tanx+cotx)`

`= sec^2x-2secxcosecx+cosec^2x`

`(secx-cosecx)^2 = sec^2x-2secxcosecx+cosec^2x`

`(1-tanx )^2 + (1-cotx )^2`

`= (1+tan^2x)+(1+cot^2x)-2(tanx+cotx)`

`= sec^2x-2secxcosecx+cosec^2x`

`= (secx-cosecx)^2 `

*So the proof is completed.*