# How to simplify $(\sin\theta-\cos\theta)^2+(\sin\theta+\cos\theta)^2$?

Simplify: $(\sin \theta − \cos \theta)^2 + (\sin \theta + \cos \theta)^2$

• 1
• 2
• $\sin^2 \theta$
• $\cos^2 \theta$

I am lost on how to do this. Help would be much appreciated.

• Do you know that $\sin^2 \theta + \cos^2 \theta =1$? If so, you have to write down the two squares and you get $2$. Sep 21, 2014 at 21:46
• See math notation guide and try asking better questions in the future.
– user147263
Sep 21, 2014 at 21:49

Hint:

1) Expand : $(a+b)^2$ and $(a-b)^2$ for all real numbers $a$ and $b$.

2) What is the value of $\sin^2(x) + \cos^2(x)$ for all real number $x$ ?

• Is it 1? I think I got it Sep 21, 2014 at 21:48
• Yes $\sin^2 (x) + \cos^2 (x) =1$. Now if you expand correctly $(a-b)^2 + (a+b)^2$, you can find the correct answer. Sep 21, 2014 at 21:50
• so 1+1 to get 2 Sep 21, 2014 at 21:52
• $\sin^2\theta+\cos^2\theta=1$? I don't get it. Sep 26, 2014 at 14:50

All you have to do is multiply it out.

$(\sin{\theta} - \cos{\theta})^{2} + (\sin{\theta} + \cos{\theta})^{2}$

$= (\sin{\theta} - \cos{\theta})(\sin{\theta} - \cos{\theta}) + (\sin{\theta} + \cos{\theta})(\sin{\theta} + \cos{\theta})$

$= \sin^{2}{\theta} - 2\sin{\theta}\cos{\theta} + \cos^{2}{\theta} + \sin^{2}{\theta} + 2\sin{\theta}\cos{\theta} + cos^{2}{\theta}$

$=\sin^{2}{\theta} + \cos^{2}{\theta} + \sin^{2}{\theta}+ cos^{2}{\theta}$

$= 1 + 1$

$= 2$

• it was a practice ACT question out of this ACT book I have, haha. Thank you though! Sep 21, 2014 at 21:54
• @tiffany Ok, I removed my suggestion about the homework from my answer. Some people like to post homework questions on here without putting any effort into understanding the problem or solution. I wanted to make sure you'd ask questions if you had any. Cheers! Sep 21, 2014 at 21:56

\begin{align} &\phantom{=}\left(\sin x-\cos x\right)^2+\left(\sin x+\cos x\right)^2\\ &=\sin^2x-2\sin x\cos x+\cos^2x+\sin^2x+2\sin x\cos x+\cos^2x\\ &=2\sin^2x+2\cos^2x\\ &=2\left(\sin^2x+\cos^2x\right)\\ &=2 \end{align}