Question

How do you simplify `tan^2 x- cot^2x`?

Answer 1

`tan^2x-cot^2x=(sin^2x)/(cos^2x)-(cos^2x)/(sin^2x)`

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

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

`=((1)(sinx+cosx)(sinx-cosx))/((1+cosx)(1-cosx)(1+sinx)(1-sinx)`

Does not appear to be able to simplify any further.

Answer 2

`tan^2x-cot^2x = -4csc(2x)cot(2x)`

Explanation:

`tan^2x-cot^2x`

`= sin^2(x)/cos^2(x) - cos^2(x)/sin^2(x)`

`= (sin^4(x)-cos^4(x))/(cos^2(x)sin^2(x))`

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

`= ((1)(-cos(2x)))/(1/4sin^2(2x))`

`=-4cos(2x)/sin^2(2x)`

`= -4csc(2x)cot(2x)`

Final Answer

Answer 3

`(sec^2 x - csc^2 x)`

Explanation:

Add 1 and (- 1) into the equation:
`f(x) = (tan^2 x + 1) - (cot^2 x + 1) =`
`f(x) = sec^2 x - csc^2 x`