Question 633784
{{{csc^4(theta) - csc^2(theta) = cot^4(theta) + cot^2(theta)}}}
There's two keys to this:<ul><li>A 4th power is the same as squared squared. So {{{csc^4(theta) = (csc^2(theta))^2}}}; and</li><li>There is an identity that connects csc squared and cot squared: {{{csc^2(x) = cot^2(x) + 1}}}</li></ul>Once you recognize these two facts, this is a farily simple problem.<br>
1. Rewrite the left side in terms of csc squared:
{{{(csc^2(theta))^2 - csc^2(theta) = cot^4(theta) + cot^2(theta)}}}
2. Replace the csc squared's with cot squared plus one:
{{{(cot^2(theta)+1)^2 - (cot^2(theta)+1) = cot^4(theta) + cot^2(theta)}}}
3. Simplify. I'll leave this part up to you with these hints:<ol><li>Use FOIL or the {{{(a+b)^2 = a^2+2ab+b^2}}} pattern to multiply out {{{(cot^2(theta)+1)^2}}}.</li><li>Combine like terms while being careful to notice the minus in front of {{{(cot^2(theta)+1)}}}.</li></ol>
If you do this correctly the left side matches the right side fairly quickly.