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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