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I am not quite sure what the question is asking here really. I understand the Lemma. Do I prove that $a^2+b^2=1$, using the given expressions for $a$ and $b$?

Thank you!

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    $\begingroup$ Please do not use pictures. $\endgroup$ – Dietrich Burde Jun 12 at 15:02
  • $\begingroup$ What is wrong with using a picture? Please explain. I did not see it anywhere in the rules that pictures are not permitted. $\endgroup$ – PomPom Jun 12 at 15:07
  • $\begingroup$ See here for some good arguments not to use pictures. $\endgroup$ – Dietrich Burde Jun 12 at 15:19
  • $\begingroup$ Thank you, fair comment! $\endgroup$ – PomPom Jun 12 at 15:23
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Yes, you need to prove that $a^2+b^2=1$, using the given expressions for $a$ and $b$. You also need to show that $a$ and $b$ are positive (you'll see then why you need $\alpha < 45^\circ$).

Then, by applying the Lemma you get the conclusion of the exercise.

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  • $\begingroup$ I am not quite sure why do I have to show that $a$ and $b$ are positive. I thought because I end up having $a=\cos2\alpha$ and $b=\sin2\alpha$ It is to show that $2\alpha$ is an acute angle. $\endgroup$ – PomPom Jun 12 at 16:25
  • $\begingroup$ @PomPom The converse of the Lemma is saying "If $a,b$ are some pair of positive numbers and ...". And the point of the problem is that you can show that there exists some $\theta$ such that ..... without using trigonometry. The entire goal of this section/chapter seems to be to introduce $\cos$ and $\sin$ and their basic properties.. This means that you also need to avoid using more deep results. $\endgroup$ – N. S. Jun 12 at 19:03

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