Question

How do you simplify `tan^2x(csc^2x-1)`?

Answer 1

By using the Trigonometric Identity : `sin^2x+cos^2x=1`

Explanation:

Divide both sides of the above identity by `sin^2x` to obtain,

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

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

`=>1+1/tan^2x=csc^2x`

`=>csc^2x-1=1/tan^2x`

Now, we are able to write : `tan^2x(csc^2x-1)" "` as `" "tan^2x(1/tan^2x)`

and the result is `color(blue)1`

Answer 2

Simplify: `tan^2 x(csc^2 x - 1)`

Explanation:

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

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