Question

prove that(cosec - cot )^2=1- cos /1+ cos​

Answer 1

$cosecθ−cotθ)2=1+cosθ1−cosθ</p><p></p><p>Step-by-step explanation:</p><p></p><p>\begin{gathered}LHS = (cosec\theta-cot\theta)^{2}\\=\big(\frac{1}{sin\theta}-\frac{cos\theta}{sin\theta }\big)^{2}\end{gathered}LHS=(cosecθ−cotθ)2=(sinθ1−sinθcosθ)2</p><p></p><p>/*</p><p></p><p>\begin{gathered}we\: know \: that \\i) cosec\theta =\frac{1}{sin\theta}\\ii)cot\theta = \frac{cos\theta}{sin\theta}\end{gathered}weknowthati)cosecθ=sinθ1ii)cotθ=sinθcosθ</p><p></p><p>\begin{gathered}= \big(\frac{1-cos\theta}{sin\theta}\big)^{2}\\=\frac{(1-cos\theta)^{2}}{(sin\theta)^{2}}\\=\frac{(1-cos\theta)^{2}}{1-cos^{2}\theta}\end{gathered}=(sinθ1−cosθ)2=(sinθ)2(1−cosθ)2=1−cos2θ(1−cosθ)2</p><p></p><p>\begin{gathered}By \: trigonometric \: identity : \\\boxed{sin^{2}\theta = 1-cos^{2}\theta}\end{gathered}Bytrigonometricidentity:sin2θ=1−cos2θ</p><p></p><p>=\frac{(1-cos\theta)(1-cos\theta)}{(1+cos\theta)(1-cos\theta)}=(1+cosθ)(1−cosθ)(1−cosθ)(1−cosθ)</p><p></p><p>After cancellation, we get</p><p></p><p>\begin{gathered}=\frac{1-cos\theta}{1+cos\theta}\\=RHS\end{gathered}=1+cosθ1−cosθ=RHS</p><p></p><p>$

Answer 2

Given to prove :-

${(cosecA - cotA )^2} = \dfrac{1 -cosA}{1 + cosA}$

To know :-

• cosecA = 1 / sinA
• cotA = cosA/sinA

Solution :-

${Take\: L.H.S}$

${(cosecA - cotA)^2}$

$( \dfrac{1}{sina} - \dfrac{cosa}{sina} ) {}^{2}$

$( \dfrac{1 - cosa}{sina}) {}^{2}$

$\dfrac{(1 - cosa) {}^{2} }{sin {}^{2}a }$

${From\:Trigonmetric\: Identities}$

${sin^2A + cos^2A = 1 }$

sin²A = 1 - cos²A

$\dfrac{(1 - cosa)(1 - cosa)}{1 - cos {}^{2}a }$

Denominator is in form of a² - b² = (a + b) ( a - b)

$\dfrac{(1 - cosa)(1 - cosa)}{(1 + cosa)(1 - cosa)}$

$\dfrac{1 - cosa}{1 + cosa}$

Hence proved !

Know more :-

Trigon metric Identities

sin²θ + cos²θ = 1

sec²θ - tan²θ = 1

csc²θ - cot²θ = 1

Trigometric relations

sinθ = 1/cscθ

cosθ = 1 /secθ

tanθ = 1/cotθ

tanθ = sinθ/cosθ

cotθ = cosθ/sinθ

Trigonmetric ratios

sinθ = opp/hyp

cosθ = adj/hyp

tanθ = opp/adj

cotθ = adj/opp

cscθ = hyp/opp

secθ = hyp/adj