Question

cos x - cot x/cos x+cot x =sin x-1/sin x+1 using identities ​

Answer 1

we know that cot x = cos x / sin x

LHS

cos x - cot x / cos x +cot x

=(cos x -(cos x / sin x))/( cos x + (cos x /sin x ))

=

$\frac{ \cos(x) \sin(x) - \cos(x) }{ \sin(x) } \times \frac{ \sin(x) }{ \cos(x) \sin(x) + \cos(x) } \\ \\ = \frac{ \cos(x) \sin(x) - \cos(x) }{ \cos(x) \sin(x) + \cos(x) } \\ \\ = \frac{ \cos(x)( \sin(x) - 1) }{ \cos(x)( \sin(x) + 1) } \\ \\ = \frac{ \sin(x) - 1 }{ \sin(x) + 1}$

= RHS