Question 271498
<pre><font size = 4 color = "indigo"><b>
{{{csc^4x - 2csc^2x + 1 = cot^4x}}}

Work with the left side only:

Factor the left side:

{{{(csc^2x-1)(csc^2-1)=""}}}

Use the identity {{{1+cot^2alpha=csc^2alpha}}} or {{{cot^2alpha=csc^2alpha-1}}}

{{{(cot^2x)(cot^2x)=""}}}

{{{cot^4x=""}}}

--------------------------------------

{{{(csc^2x)/(cot(x))  =  csc(x)*sec(x)}}}

Work with the left side only 

{{{(csc^2x)/(cot(x))=""}}}

Use the identity {{{csc(alpha)=1/(sin(alpha))}}} to
rewrite the numerator and the identity {{{cot(alpha)=(Cos(alpha))/(Sin(alpha))}}} to rewrite the denominator:


{{{

(  1/(Sin^2x) )/  (  (Cos(x)) /(Sin(x))  )

=""}}}


Write as a division:

{{{

(  1/(Sin^2x) )}}}{{{"÷"}}}{{{  (  (Cos(x)) /(Sin(x))  )=""}}}

Invert the second fraction and change the division to multiplication:

{{{

(  1/(Sin^2x) )*  (  (Sin(x)) /(Cos(x))  )=""}}}

Cancel

{{{

(  1/(Sin^cross(2)x) )*  (  (cross(Sin(x))) /(Cos(x))  )=""}}}

{{{ (1/(Sin(x)))(1/(Cos(x)))=""}}}

Use the identities {{{csc(alpha)=1/Sin(alpha)}}} and {{{sec(alpha)=1/Cos(alpha)}}}
to rewrite those factors

{{{csc(x)sec(x)=""}}}

Edwin</pre>