Simplify: $(\sin \theta − \cos \theta)^2 + (\sin \theta + \cos \theta)^2$
Answer choices:
- 1
- 2
- $ \sin^2 \theta$
- $ \cos^2 \theta$
I am lost on how to do this. Help would be much appreciated.
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2$\begingroup$ Do you know that $\sin^2 \theta + \cos^2 \theta =1$? If so, you have to write down the two squares and you get $2$. $\endgroup$– CrostulSep 21, 2014 at 21:46
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2$\begingroup$ See math notation guide and try asking better questions in the future. $\endgroup$– user147263Sep 21, 2014 at 21:49
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3 Answers
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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 $x$ ?
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All you have to do is multiply it out.
$(\sin{\theta} - \cos{\theta})^{2} + (\sin{\theta} + \cos{\theta})^{2}$
$= (\sin{\theta} - \cos{\theta})(\sin{\theta} - \cos{\theta}) + (\sin{\theta} + \cos{\theta})(\sin{\theta} + \cos{\theta})$
$= \sin^{2}{\theta} - 2\sin{\theta}\cos{\theta} + \cos^{2}{\theta} + \sin^{2}{\theta} + 2\sin{\theta}\cos{\theta} + cos^{2}{\theta}$
$=\sin^{2}{\theta} + \cos^{2}{\theta} + \sin^{2}{\theta}+ cos^{2}{\theta}$
$= 1 + 1$
$= 2$
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$\begingroup$ it was a practice ACT question out of this ACT book I have, haha. Thank you though! $\endgroup$– tiffanySep 21, 2014 at 21:54
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$\begingroup$ @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! $\endgroup$– laymanSep 21, 2014 at 21:56
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$$\begin{align} &\phantom{=}\left(\sin x-\cos x\right)^2+\left(\sin x+\cos x\right)^2\\ &=\sin^2x-2\sin x\cos x+\cos^2x+\sin^2x+2\sin x\cos x+\cos^2x\\ &=2\sin^2x+2\cos^2x\\ &=2\left(\sin^2x+\cos^2x\right)\\ &=2 \end{align}$$
