Question

$\tt \left(cosec \theta - cot \theta \right)^{2} = \dfrac{1 - cos \theta}{1 + cos \theta}$​

Answer 1

To prove :-

$\tt{(cosec\theta - cot\theta)^2=\dfrac{1-cos\theta}{1+cos\theta}}$

Proof :-

Taking LHS

$\implies\tt{LHS=(cosec\theta-cot\theta)^2}\\\\\implies\tt{LHS = {\left( \dfrac{1}{sin\theta}-\dfrac{cos\theta}{sin\theta} \right)}^2}\\\\\\\implies\tt{LHS={\left( \dfrac{1-cos\theta}{sin\theta}\right)}^2}\\\\\\ \implies\tt{LHS=\dfrac{(1-cos\theta)^2}{sin^2\theta}}\\\\\\\pink{\tt{using\:identity\:\:\:1-cos^2\theta=sin^2\theta}}\\\\\\\implies\tt{LHS=\dfrac{(1-cos\theta)(1-cos\theta)}{1-cos^2\theta}}\\\\\\\pink{\tt{using\:identity\:\:\:a^2-b^2=(a+b)(a-b)}}$

$\implies\tt{LHS=\dfrac{(1-cos\theta)(1-cos\theta)}{(1+cos\theta)(1-cos\theta)}}\\\\\\\implies\tt{LHS=\dfrac{1-cos\theta}{1+cos\theta}=RHS}$

Proved.

More trigonometric identities :-

→  Cos² θ + Sin² θ = 1

→ 1 + Tan² θ = Sec² θ

→ 1 + Cot² θ = Cosec² θ

______________________

→ sin ( 90 - A ) = cos A

→ cos ( 90 - A ) = sin A

→ tan ( 90 - A ) = cot A

→ cot ( 90 - A ) = tan A

→  sec ( 90 - A ) = cosec A

→ cosec ( 90 - A ) = sec A

Answer 2

$\rule{200}3$

To prove :- $\sf \left(cosec \theta - cot \theta \right)^{2} = \dfrac{1 - cos \theta}{1 + cos \theta}$

Proof :-

Solving L.H.S

$\green{\sf \left(cosec \theta - cot \theta \right)^{2}}$

$\dashrightarrow\:\:$ $\sf \left(\dfrac{1}{sin \theta} - \dfrac{cos \theta}{sin \theta}\right)^{2}$

$\dashrightarrow\:\:$ $\sf \left(\dfrac{1 - cos \theta}{sin \theta}\right)^{2}$

$\dashrightarrow\:\:$ $\sf \dfrac {\left(1 - cos^{2}\theta\right)}{sin^{2}\theta}$

$\dashrightarrow\:\:$ $\sf \dfrac {\left(1 - cos^{2}\theta\right)}{1 - cos^{2}\theta}$

$\red{\sf Using \:a^{2} - b^{2} = (a - b) (a + b)}$

$\red{\sf{Putting\:a = 1 \: and\:b = cos\theta}}$

$\dashrightarrow\:\:$ $\sf \dfrac {\left (1 - cos\theta \right) \left(1 + cos \theta\right)}{\left (1 - cos\theta \right) \left(1 - cos \theta\right)}$

$\dashrightarrow\:\:$ $\dfrac{1 - cos \theta}{1 + cos \theta}$

= R.H.S

Hence proved.

$\rule{200}3$

$\dagger\:$Extra shots :-

$\star$$\small\underline{\sf Trignometric\: Identies\::-}$

$\boxed{\begin{minipage}{20 em} \sf \displaystyle \bullet \cos^{2}\theta + sin^{2}\theta=1 \\\\\\ \bullet \ 1 + tan^{2}\theta=\sec^{2}\theta \\\\\\ \bullet \ 1 + cot^{2}\theta = cosec^{2}\theta \end{minipage}}$

$\rule{200}3$