Question

Answer this as soon as possible and don't attempt if you don't know. Thanks!​

Answer 1

$\largeɢɪᴠᴇɴ \: : \frac{ \cos \: A - \sin \: A + 1 }{ \cos \: A + \sin \: A - 1 } = \cosec \: A + \cot \: A$

$\largeᴛᴏ \: ᴘʀᴏᴠᴇ : LHS=RHS$

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \large\mathcal\pink{\boxed{\boxed{Solution}}}$

$LHS = \frac{ \cos \: A - \sin \: A + 1 }{ \cos \: A + \sin \: A - 1 }$

Divide and Multiply by sin A,

$\large= \frac{ \sin \: A (\cos \: A - \sin \: A + 1 )}{ \sin \: A (\cos \: A + \sin \: A - 1) }$

$\large = \frac{ \sin \: A \cos \: A - \ { \sin \: A }^{2} + \sin \: A }{ \sin \: A (\cos \: A + \sin \: A - 1) }$

$\large = \frac{ \sin \: A \cos \: A + \ { \sin \: A } - (1 - {\cos \: }^{2} A) }{ \sin \: A (\cos \: A + \sin \: A - 1) }$

$\large = \frac{ \sin \: A (\cos \: A +1) \ - (1 - {\cos \: } A) (1 + {\cos \: } A) }{ \sin \: A (\cos \: A + \sin \: A - 1) }$

$\large = \frac{ (1 + {\cos \: }A) (\sin A + \cos A - 1) }{ \sin A (\cos A + \sin \: A - 1) }$

$\large = \frac{1 + \cos A }{ \sin A}$

$\large = \frac{1}{ \sin A } + \frac{ \cos A }{ \sin A }$

$\large = cosec A + \cot A$

$=RHS$

Therefore LHS=RHS

Hence Proved..

$\large\sf\underline{\orange{Hope \:  It  \: Helps \:  You!}}$