How do you prove $\frac{1 - cos (x)}{sin (x)} = csc (x) - cot (x)$ $(1-cos(x)) / sin(x) = csc(x) - cot(x)$ ?

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How do you prove $\frac{1 - cos (x)}{sin (x)} = csc (x) - cot (x)$ $(1-cos(x)) / sin(x) = csc(x) - cot(x)$ ?

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Answer 1 · 2016-04-30T00:46:59

Recall the following identities:

$1 . csc x = \frac{1}{sin x}$ $1. color(red)(cscx=1/sinx)$

$2 . cot x = \frac{cos x}{sin x}$ $2. color(blue)(cotx=cosx/sinx)$

Given,

$\frac{1 - cos x}{sin x} = csc x - cot x$ $(1-cosx)/sinx=cscx-cotx$

Simplify the right side.

$csc x - cot x$ $color(red)(cscx)-color(blue)(cotx)$

$= \frac{1}{sin x} - \frac{cos x}{sin x}$ $=color(red)(1/sinx)-color(blue)(cosx/sinx)$

$= \frac{1 - cos x}{sin x}$ $=(1-cosx)/sinx$

$:.$ , LS $=$ RS.

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How do you prove $\frac{1 - cos (x)}{sin (x)} = csc (x) - cot (x)$ $(1-cos(x)) / sin(x) = csc(x) - cot(x)$ ?

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