Establish the identity.&nbsp;1-cosθ1+cosθ=(cscθ-cotθ)2

Given:1−cos⁡θ1+cos⁡θ=(csc⁡θ−cot⁡θ)2RHS(csc⁡θ−cot⁡θ)2⇒&nbsp;(1)csc⁡θ=1sin⁡θcot⁡θ=cos⁡θsin⁡θ&nbsp;and&nbsp;(a−b)2=a2−2ab+b2from (1)(csc⁡θ−cot⁡θ)2=(1sin⁡θ−cos⁡θsin⁡θ)2Apply formula(1sin⁡θ)2−2(1sin⁡θ)(cos⁡θsin⁡θ)+(cos⁡θsin⁡θ)2(csc⁡θ−cot⁡θ)2=1sin2θ−2cos⁡θsin2θ+cos2θsin2θ(cscθ−cotθ)2⇒(1)=1−2cos⁡θ+cos2θsin2θ=(1−cos⁡θ)2sin2θ∵cos2θ+sin2θ=1(cscθ−cotθ)2⇒sin2θ=1−cos2θ=(1−cos⁡θ)(1−cos⁡θ)(1−cos2θ)(cscθ−cotθ)2⇒(2)=(1−cos⁡θ)(1−cos⁡θ)(1−cos⁡θ)(1+cos⁡θ)=1−cos⁡θ1+cos⁡θ(cscθ−cotθ)2⇒(3)from (1)(2)(3)(cscθ-cotθ)2=1sin2θ-2cosθsin2θ+cos2θsin2θ=(1-cosθ)21-cos2θ=1-cosθ1+cosθ

Expert answers to "Establish the identity.1-cosθ1+cosθ=(cscθ-cotθ)2"

High School
Geometry
Trigonometry
Trigonometric equations and identities

puntgewelb5

Answered question

2021-12-02

Establish the identity. $\frac{1 - c o s θ}{1 + c o s θ} = (c s c θ - c o t θ)^{2}$

Answer & Explanation

Helen Rodriguez

Beginner2021-12-03Added 9 answers

Given:
$\frac{1 - \cos θ}{1 + \cos θ} = {(\csc θ - \cot θ)}^{2}$
RHS $(\csc θ - \cot θ)^{2} \Rightarrow$ (1)
$\csc θ = \frac{1}{\sin θ} \cot θ = \frac{\cos θ}{\sin θ}$
$and {(a - b)}^{2} = a^{2} - 2 a b + b^{2}$
from (1) ${(\csc θ - \cot θ)}^{2} = {(\frac{1}{\sin θ} - \frac{\cos θ}{\sin θ})}^{2}$
Apply formula
${(\frac{1}{\sin θ})}^{2} - 2 (\frac{1}{\sin θ}) (\frac{\cos θ}{\sin θ}) + {(\frac{\cos θ}{\sin θ})}^{2}$
${(\csc θ - \cot θ)}^{2} = \frac{1}{\sin^{2} θ} - \frac{2 \cos θ}{\sin^{2} θ} + \frac{\cos^{2} θ}{\sin^{2} θ} (c s c θ - c o t θ)^{2} \Rightarrow (1)$
$= \frac{1 - 2 \cos θ + \cos^{2} θ}{\sin^{2} θ}$
$= \frac{{(1 - \cos θ)}^{2}}{\sin^{2} θ}$
$∵ \cos^{2} θ + \sin^{2} θ = 1 (c s c θ - c o t θ)^{2} \Rightarrow \sin^{2} θ = 1 - \cos^{2} θ$
$= \frac{(1 - \cos θ) (1 - \cos θ)}{(1 - \cos^{2} θ)} (c s c θ - c o t θ)^{2} \Rightarrow (2)$
$= \frac{(1 - \cos θ) (1 - \cos θ)}{(1 - \cos θ) (1 + \cos θ)}$
$= \frac{1 - \cos θ}{1 + \cos θ} (c s c θ - c o t θ)^{2} \Rightarrow (3)$
from (1)(2)(3)
$(c s c θ - c o t θ)^{2} = \frac{1}{s i n^{2} θ} - \frac{2 c o s θ}{s i n^{2} θ} + \frac{c o s^{2} θ}{s i n^{2} θ}$

$= \frac{(1 - c o s θ)^{2}}{1 - c o s^{2} θ} = \frac{1 - c o s θ}{1 + c o s θ}$

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