Question 1151556
<pre>{{{cos^6(x)=-2+3cos^2(x)+3sin^4(x)-sin^6(x)}}}

{{{(cos^2(x)^"")^3=-2+3cos^2(x)+3sin^4(x)-sin^6(x)}}}

Substitute 1-sin²(x) for cos²(x)

{{{(1-sin^2(x)^"")^3=-2+3(1-sin^2(x)^"")+3sin^4(x)-sin^6(x)}}}

{{{(1-sin^2(x)^"")^3=-2+3(1-sin^2(x)^"")+3(sin^2(x)^"")^2-(sin^2(x)^"")^3}}}

To uncomplicate matters, let y = sin²(x)

{{{(1-y^"")^3=-2+3(1-y^"")+3(y^"")^2-(y^"")^3}}}

Simplify

{{{1-3y+3y^2+y^3=-2+3-3y+3y^2-y^3}}}

Simplify further

{{{1-3y+3y^2-y^3=1-3y+3y^2-y^3}}}

Both sides are identical, so the problem is proved.
It's OK to work with both sides of an identity as long
as you don't change the value of either side, by adding,
subtracting, multiplying, or dividing to, from, or by both
sides.

Edwin</pre>