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find the derivative of g(x)=(tan^2)(x)-(csc^2)(x)

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`g(x)= tan^2 x - csc^2 x `

First, we know that

`(tanx)' = sec^2 x `

`(cscx)' = -cscx*cotx`

Now we will differentiate

`==gt g'(x)= 2tanx(tanx)' - 2cscx (csc)'`

` ==gt g'(x)= 2tanx(sec^2 x)- 2cscx(-cscx*cotx) `

`==gt g'(x)= 2tanx*sec^2 x + 2csc^2 x cotx `

`==gt g'(x)= 2(sinx)/(cosx)(1/(cos^2 x)) + 2(1/(sin^2 x) (cosx)/(sinx)) `

`==gt g'(x)= (2sinx)/(cos^3x) + (2cosx)/(sin^3 x) `

`==gt g'(x)= (2sin^4 x + 2cos^4 x)/(sinxcosx)^3 `

`==gt g'(x)= 2((sin^2 x + cos^2 x)^2 -2(sinxcosx)^2)/(sinxcosx)^3`

` ==gt g'(x)= 2( 1 -2(sinxcosx)^2)/(sinxcosx)^3 `

`==gt g'(x)= 2(secx*cscx)^3 - 4secx*cscx`

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