Came up with a second personalized proof (is it correct)

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The discussion centers on the proof of the identity sin(90 + theta) = cos(theta). Participants debate the construction of angles in a triangle, specifically questioning the relationship between angles COD and ODE. There is confusion regarding whether angle COD should be expressed as 45 - (theta/2) and the properties of isosceles triangles in relation to angle bisectors. Clarifications are made about the perpendicularity of segments OD and AE, leading to revised steps in the proof. The conversation emphasizes the importance of accurately demonstrating angle relationships in geometric proofs.
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proof of sin(90 + theta) = cos(theta)
 

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Well, I wouldn't say "by construction angle COD is also theta". Since OD is perpendicular to AE, the only thing that is true "by construction" is that angle ODE and ODA are right angles. Now, you can they show that COD is theta by using the fact that the two angles in a right triangle are complementary- but you should show that.
 
Wait. Am I missing something here. Shouldn't COD be 45-(theta/2) ?

AOE is an isosceles triangle. The perpendicular is also the angle bisector, right? Or have I understoond something wrong here?
 
Sorry.. after I looked at it... segment OD is not perp to segment AE.. and it doesn't have to be. I came up with a diff steps which I edited in the picture that I attached..
 

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