# prove that (tanx+cotx)^4=sec^4x csc^4x

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The identity `(tan x + cot x)^4 = sec^4 x*csc^4 x` has to be proved.

`(tan x + cot x)^4`

`tan x = sinx/cos x` and `cot x = cos x/sin x`

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

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

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

=> `1/(sin x*cos x)^4`

=> `1/(sin^4 x*cos^4x)`

=> `sec^4x*cosec^4x`

**This proves `(tan x + cot x)^4 = sec^4 x*csc^4 x`**

(tan x + cot x)^4 = [(sinx/cosx) + (cosx/sinx)]^4

= [(sin^2 x + cos^2 x)/(sinx cosx)]^4

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

= [1/(sinx cosx)]^4

= (csc x . sec x)^4

= csc^4 x sec^4 x

(tan x+cotx)^4 = (sinx/cosx + cosx/sinx)^4

= (sin^2 x + cos^x/(sinx cosx))^4

=(1/cosx. sinx)^4 [since sin^2x + cos^x= 1]

= (1/sinx . 1/cosx)^4

=(cosec x sec x)^4

= cosex ^4 x . sec ^4x

**Sources:**