Question

1+cosA-sin²A/sinA(1+cosA)=cotA proove that

Answer 1

We have to prove that :

$\frac{1 + \cos\alpha - { \sin }^{2} \alpha }{ \sin\alpha (1 + \cos\alpha )} = \cot\alpha$

As we know that :

${ \sin}^{2} \alpha + { \cos}^{2} \alpha = 1 \\ \\ = > 1 - { \sin }^{2} \alpha = { \cos}^{2} \alpha$

On taking L.H.S. :

$\frac{1 + \cos \alpha - { \sin }^{2} \alpha }{ \sin \alpha (1 + \cos \alpha ) } \\ \\ = > \frac{1 - { \sin }^{2} \alpha + \cos\alpha }{ \sin \alpha (1 + \cos\alpha ) } \\ \\ = > \frac{ { \cos}^{2} \alpha + \cos\alpha }{ \sin \alpha (1 + \cos \alpha )} \\ \\ = > \frac{ \cos\alpha ( \cos \alpha + 1) }{ \sin \alpha(1 + \cos \alpha ) } \\ \\ = > \frac{ \cos \alpha }{ \sin \alpha } \\ \\ = > \cot\alpha$

L.H.S. = R.H.S.

HENCE PROVED