Question

Prove the Following :

$\sqrt{ \frac{1 - \cos \: a }{1 + cos \: a} } \: = cosec \: a \: - cot \: a$
​

Answer 1

Answer :

Given To Prove

$\longrightarrow \: \mathsf{\sqrt{ \dfrac{1 - \cos \: a }{1 + \cos \: a} } \: = \cosec \: a \: - \cot \: a}$

Proof

∆ Taking LHS

$\longrightarrow \: \mathsf{\sqrt{ \dfrac{1 - \cos \: a }{1 + \cos \: a} }}$

Rationalising the denominator we have

$\longrightarrow \: \mathsf{\sqrt{ \dfrac{1 - \cos \: a }{1 + \cos \:a} } }$

$\longrightarrow \: \mathsf{\sqrt{ \dfrac{1 - \cos \: a }{1 + \cos \:a} \times \dfrac{1 - \cos \: a}{1 - \cos a} }}$

Now we know

$\boxed{\boxed{\mathsf{ (a + b) (a - b) = a^2 - b^2}}}$

$\longrightarrow \: \mathsf{\sqrt{ \dfrac{ {(1 - \cos a) }^{2} }{ {(1)}^{2} - { \cos }^{2} \: a} }}$

Now We have

$\boxed{\boxed{\mathsf{\bigstar \: 1 - {\cos}^{2} a = {\sin}^2 a}}}$

$\longrightarrow \: \mathsf{\sqrt{ \dfrac{ {(1 - \cos a )}^{2} }{ { \sin }^{2} a }}}$

$\longrightarrow \: \mathsf{ \dfrac{ 1 - \cos a }{ \sin a }}$

$\longrightarrow \: \mathsf{ \dfrac{ 1 }{ \sin a } - \dfrac{ \cos a}{ \sin a} }$

Now we know

$\boxed{ \boxed{\mathsf{\bigstar \: \dfrac{ 1 }{ \sin a } = \cosec a \: \: \& \: \: \dfrac{ \cos a}{ \sin a} = \cot a}}}$

Therefore We have

→ LHS →

$\implies \mathsf{\cosec a \: - \cot a}$

→ RHS →

$\implies \mathsf{\cosec a \: - \cot a}$

Therefore We Have

$\large \mathcal{\bigstar \:LHS \: = \: RHS}$

$\rule{300}{1}$

$\underline{\underline{\mathsf{\star \: More \: To \: Know \: \star}}}$

$\longrightarrow \: \underline{Pythagorean\: Identies}$

$\mathsf{\star \: {\sin}^{2} \theta + {\cos}^{2} \theta = 1}$

$\mathsf{\star \: {\tan}^{2} + 1 = {\sec}^{2} \theta }$

$\mathsf{\star \: {\cot}^{2} + 1 = {\cosec}^{2} \theta }$

Answer 2

${\bold{\underline{\underline{Answer:}}}}$

$\implies \: <strong>To</strong> \:<strong>Prove</strong> = \sqrt{ \frac{1 - cos \: a \: }{1 + cos \: a} } = cosec \: a - cot \: a$

Proof :

Taking L.H.S we get

$\implies \sqrt{ \frac{1 - \cos \: a }{1 + \cos \: a } }$

$\implies$ Multiply Numerator and denominator by (1 – Cos a)

$\implies \sqrt{ \frac{1 - cos \: a}{ 1 + \cos \: a } }$

$\implies \sqrt{ \frac{1 - \cos \: a }{1 + cos \: a} \times \frac{1 - \cos \: a }{1 - \cos \: a } }$

since ( a + b ) ( a - b ) = a²2 - b²

$\sqrt{ \frac{(1 - cos \: a^{2}) }{(1^{2}) - cos^{2}\:a}}$

$\implies$ We know that (1–Cos²a = Sin² a)

$\sqrt{ \frac{1 - \cos \:a }{ \sin ^{2} a }} = \frac{1 - cos \: a}{sin \: a}$

$\implies \frac{1}{ \sin} - \frac{ \cos }{sin}$

$\implies \: cosec \: a - cot \: a$

$\implies$ Hence Proved.

_________________________________

$\frac{1}{sin \: a} = cosec \: a$

$\frac{ \cos \: a }{ \sin \: a } = cot \: a$