csc^2 x-1/csc^2x=cos^2x

Please explain step by step

First of all the identity you wrote is incorrect, because:


csc^2(x) - 1/csc^2x = csc^2x \ - \ \frac 1 {csc^2(x)

and this does not equal \cos^2x

However, I suspect what you meant is this:

(csc^2(x) - 1)/csc^2x =cos^2x

It is imperative that you use parentheses properly!

As usual with these types of problems, start by replacing any functions of tan, cot, sec, and csc with their sin and cos equivalents. In this case you replace csc(x) with 1/sin(x) and the left hand side becomes


\frac {(1/ \sin^2 x) - 1}{1/ \sin^2x}


Then simply - multiply through by \sin^2x and apply the identity \sin^2x + \cos^2x = 1.