So I am having a hard time with this math question:

$$1-2\sin ^2(x)+\sin^4(x)$$

The prompt is to factor the trigonometric equation and simplify. I get it simplified to a point but the answer I keep getting is:

$$(1 + \sin^2(x))(\cos^2(x))$$

For some reason the answer key says the answer is $\cos^4(x)$ but I have no idea how they are getting to that conclusion.


Method-1: $$1-2\sin^2 x+\sin^4 x=(1)^2-2(1)(\sin^2 x)+(\sin^2 x)^2$$ By applying $a^2-2ab+b^2=(a-b)^2$ $$=(1-\sin^2 x)^2$$ $$=(\cos^2 x)^2$$ $$=\color{red}{\cos^4 x}$$

Method-2: $$1-2\sin^2 x+\sin^4 x=1-\sin^2 x-\sin^2 x+\sin^4 x$$ $$=(1-\sin^2 x)-\sin^2 x(1-\sin^2 x)$$ $$=\cos^2 x-\sin^2 x(\cos^2 x)$$ $$=\cos^2 x(1-\sin^2 x)$$ $$=\cos^2 x(\cos^2 x)$$ $$=\color{red}{\cos^4 x}$$


Hint: $1-2\sin^2(x)+\sin^4(x)=(1-\sin^2(x))^2$

  • $\begingroup$ This is obviously the part that the asker was stuck with, how does this help them understand how to solve this problem? Giving a hint like this doesn't teach them anything, it just let's them find the answer to this question specifically. $\endgroup$ – David Sep 14 '15 at 3:28

Let me try!

Let $\sin^2 x = u$

then, $$1 - 2\sin^2 x + \sin^4 x= 1 - 2u + u^2 =(1 - u)^2$$

Substituting the trig function back in:

$$(1 - u)^2 = (1 - \sin^2 x)^2 = (cos^2 x)^2 $$ $$=\cos^4 x$$


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