How do you prove csc^4x-cot^4x=csc^2x+cot^2x?

2 Answers
May 25, 2015

Left side:

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

= [(1- cos^2 x)(1 + cos^2 x)]/(sin^4 x) = ((sin^2 x)(1 + cos ^2 x))/(sin^4 x)
= (1 + cos^2x)/sin^2 x = 1/(sin^2 x)+ (cos^2 x)/(sin^2 x) =
= csc^2 x + cot^2 x

Jan 26, 2017

Using the Identity ; csc^2y-cot^2y=1,

The L.H.S. =csc^4x-cot^4x=(csc^2x-cot^2x)(csc^2x+cot^2x)
=csc^2x+cot^2x=the R.H.S.