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I have the following expression

$\cos^2(s) + 2\sin^2(s)$

and need to show that it is equivalent to the following expression

$1 + \sin^2(s)$

I have no idea where to start, however. I'm familiar with the use of Euler's Formula to derive the angle addition and subtraction formulas--and from there, the double and half angle formulas.

Where do we start to go from the first statement above to the second? Can you tie the trig identities you use back to Euler's Formula?

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    $\begingroup$ Do you know the result of $\sin^2 x + \cos^2 x $? $\endgroup$ – Fabian Apr 12 '16 at 11:49
  • $\begingroup$ And in LaTeX don't use \ to make spacing and use \cos and \sin upright. $\endgroup$ – user21820 Apr 12 '16 at 11:51
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$$\cos^2(x) + 2\sin^2(x) = \cos^2(x) + \underbrace{\sin^2(x) + \sin^2(x)}_{2\sin^2(x)}$$

Now, since $\cos^2(x) + \sin^2(x) = 1$

you easily get

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

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    $\begingroup$ A complete solution leaves little for the asker to try or figure out on their own. $\endgroup$ – user21820 Apr 12 '16 at 11:53
  • $\begingroup$ But is exactly what I'm looking for. Thank you! I learned something. +1 $\endgroup$ – StudentsTea Apr 12 '16 at 11:54
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    $\begingroup$ @user21820 A complete solution sometimes opens the mind of the asker more than leaving him to guess and try for hours. $\endgroup$ – Von Neumann Apr 12 '16 at 12:03
  • $\begingroup$ @user1739757 You're welcome!! Never forget about those "tricks" and important identities! $\endgroup$ – Von Neumann Apr 12 '16 at 12:03
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    $\begingroup$ @HeliumAtom: In my experience as a teacher, that's simply not true. For homework, complete solutions are not desirable. Of course guessing and trying should always be kept at the limit of the student's ability, not further. Anyway, do what you like. If you teach long enough you'll perhaps change your mind. $\endgroup$ – user21820 Apr 12 '16 at 12:13
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Always remember $\cos,\sin$ as the $x,y$ coordinates of a point on the unit circle. Then Pythagoras theorem tells you that the sum of their squares is $1$. And that solves a lot of such identities.

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  • $\begingroup$ This kind of back-and-forth dialog is probably best done in chat. $\endgroup$ – StudentsTea Apr 12 '16 at 11:55
  • $\begingroup$ @user1739757: That's very true, but usually it's better to show the crux of the matter rather than spoon-feed. Most students never learn to do mathematics on their own because they are never encouraged to try. $\endgroup$ – user21820 Apr 12 '16 at 11:57

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