Question

sinA/1-cosA=cosecA+cotA​

Answer 1

To prove :

$\star \: \dfrac{ \sin \theta}{1 - \cos \theta} = \cosec \theta \: + \cot \theta$

Solution :

• LHS

$\implies \: \dfrac{ \sin\theta}{1 - \cos\theta}$

Rationalise the denominator :

$\implies \: \dfrac{ \sin\theta}{1 - \cos\theta} \times \dfrac{1 + \cos \: \theta }{1 + \cos\theta} \\ \\ \implies \: \dfrac{ (\sin \theta)(1 + \cos \theta)}{ {1}^{2} - { \cos}^{2}\theta } \\ \\ \implies \: \dfrac{ ( \sin\theta)(1 + \cos \theta)}{ { \sin}^{2} \theta} \\ \\ \implies \: \dfrac{ \cancel{\sin \theta}}{ \cancel{{ \sin}^{2 } \theta} } + \dfrac{ \cancel{ \sin \theta} \cos \theta} { \cancel{ { \sin}^{2} \theta} } \\ \\ \implies \: \dfrac{1}{ \sin \theta} + \dfrac{ \cos\theta}{ \sin\theta} \\ \\ \implies \: \cosec\theta + \cot \: \theta$

LHS = RHS

PROVED.

Formula used :

• sin² @ + cos²@ = 1

• sin @ = 1/ cosec@

• cot@ = cos@/ sin@