# ChallengeSimplify a sum

#### anemone

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

MarkFL

#### MarkFL

##### La Villa Strangiato
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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:

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

anemone

#### anemone

##### Paris la ville de l'amour
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Very good, Mark!

The following is the second part of the challenge:

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

Last edited:
MarkFL

#### MarkFL

##### La Villa Strangiato
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If we use a co-function identity, then the product $$P$$ may be written:

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

anemone

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