Simplify
$\cos^2(v) + \sin^2(-v)$
Attempt:
$\cos^2(v) + \sin^2(-v) = 1$
$\cos^2v + \sin v = 1$
$\sqrt{\cos^2v} + \sin v = 1$
$\cos v + \sin v = 1$
The answer in the book is $1$ after the simplification.
I am using this table:
AI:
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$\begingroup$ Your attempt gives 1, just as you say the book does, so what's the issue? $\endgroup$Henrik supports the community– Henrik supports the community2024-09-03 09:42:43 +00:00Commented Sep 3, 2024 at 9:42
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$\begingroup$ @Henriksupportsthecommunity, since I am still new to this, and currently struggling, I check against the internet to verify with others that know for sure, if my steps are correct here or not. I am studying this type of math from home. Remotely. Meaning, it takes days for a teacher to answer your questions. So, after a while, I post here. $\endgroup$Alix Blaine– Alix Blaine2024-09-03 09:44:52 +00:00Commented Sep 3, 2024 at 9:44
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$\begingroup$ Then say in the post that you are looking for verification of your solution. $\endgroup$Henrik supports the community– Henrik supports the community2024-09-03 10:32:09 +00:00Commented Sep 3, 2024 at 10:32
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2 Answers
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Note that $\sin^2(-x)$ is an even function, in fact since $\sin x$ is odd, you have: $$\sin^2(-x)=\sin(-x)\cdot\sin(-x)=(-\sin x)\cdot(-\sin x)=\sin^2(x) $$ Hence the answer is $1$.
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1$\begingroup$ @AlixBlaine Sorry, I don't understand your question $\endgroup$Sine of the Time– Sine of the Time2024-09-03 09:47:10 +00:00Commented Sep 3, 2024 at 9:47
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1$\begingroup$ @AlixBlaine Why in the second step $\sin^2(-v)$ become $\sin v$? $\endgroup$Sine of the Time– Sine of the Time2024-09-03 09:51:47 +00:00Commented Sep 3, 2024 at 9:51
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1$\begingroup$ You can't consider the square root of $\cos^2 x$ while leaving the other terms. $\endgroup$Sine of the Time– Sine of the Time2024-09-03 09:53:59 +00:00Commented Sep 3, 2024 at 9:53
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1$\begingroup$ @AlixBlaine No need to be sorry $\endgroup$Sine of the Time– Sine of the Time2024-09-03 09:55:52 +00:00Commented Sep 3, 2024 at 9:55
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1$\begingroup$ @SineoftheTime, thanks a lot. Salam alaykum!!! $\endgroup$Alix Blaine– Alix Blaine2024-09-03 10:00:08 +00:00Commented Sep 3, 2024 at 10:00
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Note that $\sin^2 (-x)$ means $\sin(-x)$ has been raised to the second power i.e.
$\sin^2 (-x)$ = $\ [sin(-x)]^2$ = $\ [-sin(x)]^2$ = $\sin^2 (x)$
Hence, $\ cos^2(x) + sin^2 (-x)$ => $\ cos^2(x) + sin^2 (x)$
Which gives 1 , the desired result.
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$\begingroup$ How is this different from my answer? $\endgroup$Sine of the Time– Sine of the Time2024-09-03 09:56:02 +00:00Commented Sep 3, 2024 at 9:56
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$\begingroup$ Couldn't see your answer (glitch prolly). Sorry. $\endgroup$Krishang Rana– Krishang Rana2024-09-03 09:57:23 +00:00Commented Sep 3, 2024 at 9:57
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$\begingroup$ @KrishangRana, thanks for your attempt. +1 from me. $\endgroup$Alix Blaine– Alix Blaine2024-09-03 09:57:52 +00:00Commented Sep 3, 2024 at 9:57