Use the following notation `alpha` =angle.
You need to prove the identity:
You need to remember that tan alpha = sin alpha/cos alpha and cot alpha = cos alpha/sin alpha
`(1-sin^2 alpha)/(cos^2 alpha)/(1-cos^2 alpha)/(sin^2 alpha)=1-sec^2alpha`
The left side seems more complicated, hence you will expand such that:
`((cos^2 alpha-sin^2 alpha)/(cos^2 alpha))/((sin^2 alpha-cos^2 alpha)/(sin^2 alpha))`
Reducing by `cos^2 alpha-sin^2 alpha` yields:
`(1/(cos^2 alpha))/((-1)/(sin^2 alpha)) = - tan^2 alpha`
There is not much that you may do to the left side, hence you may move to the right side and expand such that:
`1-sec^2alpha = 1 - (1 + tan^2 alpha)`
Opening the brackets yields:
`1-sec^2alpha = - tan^2 alpha`
Since the both sides of the last equation are equal, hence the original identity is checked.