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# Help geometry

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PROVE

(1+cos2θ)/sin2θ = cotθ

May 14, 2018

### 1+0 Answers

#1
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$$\dfrac{1+\cos2\theta}{\sin2\theta}=\cot\theta$$

Let's turn the left side into the right side.

$$\phantom{=\,}\ \dfrac{1+\cos2\theta}{\sin2\theta}$$

First let's use the double-angle identity for sin:    $$\sin2θ=2 \sinθ \cosθ$$

$$=\,\dfrac{1+\cos2\theta}{2\sin\theta\cos\theta}$$

Now let's use this double-angle identity for cos:  $$\cos2\theta=2\cos^2\theta-1$$

$$=\,\dfrac{1+2\cos^2\theta-1}{2\sin\theta\cos\theta}$$

Add together the 1 and the -1  in the numerator to get 0 .

$$=\,\dfrac{2\cos^2\theta}{2\sin\theta\cos\theta}$$

Divide the numerator and denominator by  2 .

$$=\,\dfrac{\cos^2\theta}{\sin\theta\cos\theta}$$

Divide the numerator and denominator by  $$\cos\theta$$  .

$$=\,\dfrac{\cos\theta}{\sin\theta}$$

And by the quotient identity for cotangent,  $$\cot\theta=\frac{\cos\theta}{\sin\theta}$$

$$=\,\cot\theta$$

.
May 15, 2018

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