Question

prove that L.H.S = R.H.S​

Answer 1

$\Large{\underline{\underline{\mathfrak{\bf{Question}}}}}$

To prove:-

$\sf{\orange{\:\dfrac{(1+\sin A)}{(\csc A-\cot A)}\:-\:\dfrac{(1-\sin A)}{(\csc A+\cot A)}}} \\ \\ \sf{\orange{\:=\:2(1+\cot A)}}$

$\Large{\underline{\underline{\mathfrak{\bf{Solution}}}}}$

$\Large{\underline{\mathfrak{\bf{\pink{Take\:L.H.S.}}}}}$

$\mathfrak{\bf{\:\dfrac{(1+\sin A)}{(\csc A-\cot A)}\:-\:\dfrac{(1-\sin A)}{(\csc A+\cot A)}}}$

$\small\sf{\green{\:\:\:\:\:\:\:\left(\dfrac{\cos A}{\sin A}\:=\:\cot A\right)}} \\ \\ \small\sf{\green{\:\:\:\:\:\:\:\left(\dfrac{1}{\sin A}\:=\:\csc A\right)}}$

$\mapsto\sf{\:\dfrac{(1+\sin A)}{(\dfrac{1}{\sin A}-\dfrac{\cos A}{\sin A})}\:-\:\dfrac{(1-\sin A)}{(\dfrac{1}{\sin A}+\dfrac{\cos A}{\sin A})}}$

$\mapsto\sf{\:\dfrac{\sin A(1+\sin A)}{(1-\cos A)}\:-\:\dfrac{\sin A(1-\sin A)}{(1+\cos A)}}$

$\mapsto\sf{\:\dfrac{\sin A(1+\cos A)(1+\sin A)-\sin A(1-\cos A)(1-\sin A)}{(1-\cos^2 A)}}$

$\small\sf{\green{\:\:\:\:\:\:\:(1-\cos^2 A)\:=\:\sin^2 A}}$

$\mapsto\sf{\:\dfrac{\sin A(1+\sin A+\cos A.\sin A+\cos A)-\sin A(1-\sin A-\cos A+\cos A.\sin A)}{\sin^2 A}}$

$\mapsto\sf{\:\dfrac{\sin A(1+\sin A+\cos A.\sin A+\cos A-1+\sin A+\cos A-\cos A.\sin A)}{\sin^2 A}}$

$\mapsto\sf{\:\dfrac{\sin A(2\sin A+2\cos A)}{(\sin A.\sin A)}}$

$\mapsto\sf{\:\dfrac{2(\sin A+\cos A)}{\sin A}}$

$\mapsto\sf{\:2\left(\dfrac{\sin A}{\sin A}+\dfrac{\cos A}{\sin A}\right)}$

$\mapsto\sf{\pink{\:2(1+\cot A)}} \\ \\ \mathfrak{\bf{\:\:\:\:\:\:\:=\:R.H.S.}}$

That's proved.

Answer 2

Hi got question ok make point bro man thank you brainylist on give known