Question

prove csc^4x-2csc^2x+1=cot^4x

Answer 1

try chaning the squares into their pythagorean id's and play with that

Answer 2

Show:
\[\csc^4(x)-2\csc^2(x)+1=\cot^4(x)\]
Writing it in terms of sines and cosines we see:
\[\left( \frac{1}{\sin(x)}\right)^4-2 \left(\frac{1}{\sin(x)} \right)^2+1=\left( \frac{\cos(x)}{\sin(x)} \right)^4\]
=
\[\frac{1}{(1-\cos^2(x))^2}-\frac{2}{1-\cos^2(x)}+\frac{(1-\cos^2(x))^2 }{(1-\cos^2(x))^2 }=\frac{1-2(1-\cos^2(x))+(1-\cos^2(x))^2 }{(1-\cos^2(x))^2}\]
=
\[\frac{1-2+2\cos^2(x)+(1-\cos^2(x))^2}{(\sin^2(x))^2}=\frac{-1+2\cos^2(x)+1+\cos^4(x)-2\cos^2(x)}{\sin^4(x)}\]
\[=\frac{\cos^4(x)}{\sin^4(x)}=\cot^4(x). Q.E.D.\]