Challenge Simplify a sum

anemone

Paris la ville de l'amour
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#1
Simplify \(\displaystyle \tan x \cos 1 ^\circ + \sin 1^\circ\).
 
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MarkFL

La Villa Strangiato
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#2
I would write:

\(\displaystyle \tan(x)\cos\left(1^{\circ}\right)+\sin\left(1^{\circ}\right)=\frac{\sin(x)\cos\left(1^{\circ}\right)+\cos(x)\sin\left(1^{\circ}\right)}{\cos(x)}\)

Applying the angle-sum identity for sine, we may write:

\(\displaystyle \tan(x)\cos\left(1^{\circ}\right)+\sin\left(1^{\circ}\right)=\frac{\sin\left(x+1^{\circ}\right)}{\cos(x)}\)
 
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anemone

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#3
Very good, Mark!

The following is the second part of the challenge:

Evaluate \(\displaystyle \prod_{x=0}^{89} (\tan x \cos 1 ^\circ + \sin 1^\circ) \)
 
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MarkFL

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#4
If we use a co-function identity, then the product \(P\) may be written:

\(\displaystyle P=\prod_{x=0}^{89}\left(\frac{\cos\left((89-x)^{\circ})\right)}{\cos(x^{\circ})}\right)=1\)
 
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