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How do you prove the identity: `(tanx+cotx)^2=sec^2x csc^2x ?` ``

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Tushar Chandra eNotes educator | Certified Educator

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We have to prove: `(tan x + cot x)^2 = sec^2 x + cosec^2x`

`sec^2x + cosec^2x`

=> `1/(cos^2 x) + 1/(sin^2 x)`

=> `(sin^2x + cos^2x)/(sin^2x*cos^2x)`

=> `1/(sin^2x*cos^2x)` ...(1)

`(tan x + cot x)^2`

=> `tan^2 x + cot^2x + 2*tan x*cot x`

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

=> `(sin^4x + cos^4x)/(cos^2x*sin^2x) + 2`

=> `((sin^2x +...

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hala718 eNotes educator | Certified Educator

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chaobas | Student

(tan x+cotx)^2 = (sinx/cosx + cosx/sinx)^2 

                       = (sin^2 x + cos^x/(sinx cosx))^2

                       =(1/cosx. sinx)^2  [since sin^2x + cos^x= 1]

                       = (1/sinx . 1/cosx)^2

                        =(cosec x   sec x)^2

                         = cosex ^2 x . sec ^2x