Math, asked by SamyekShakya, 11 months ago

please prove the following identity

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Answered by Anonymous
104

AnswEr :

To Prove :

\small \sf{ \dfrac{ \cot(A) + \csc(A) - 1 }{ \cot(A) - \csc(A) + 1 } = \dfrac{ 1 + \cos(A) }{ \sin(A) } }

Proof :

\Longrightarrow \rm{\dfrac{ \cot(A) + \csc(A) - 1 }{ \cot(A) - \csc(A) + 1 }}

⠀⠀⠀⠀⋆ (csc²A - cot²A) = 1

\Longrightarrow \rm{\dfrac{ \cot(A) + \csc(A) - (\csc^{2} (A) - \cot^{2} (A) )}{ \cot(A) - \csc(A) + 1 }}

\Longrightarrow \rm{\dfrac{ \cot(A) + \csc(A) - (\csc (A) - \cot (A) )(\csc (A) + \cot (A) )}{ \cot(A) - \csc(A) + 1 }}

⠀⠀⠀⠀⋆ taking (cotA + cscA) Common

\Longrightarrow \rm{\dfrac{ (1 - (\csc (A) - \cot (A) )(\csc (A) + \cot (A) )}{ \cot(A) - \csc(A) + 1 }}

\Longrightarrow \rm{\dfrac{ \cancel{(1 - \csc (A) + \cot (A) )}(\csc (A) + \cot (A) )} {\cancel{1 - \csc(A) + \cot(A) }}}

\Longrightarrow \rm{ \csc(A) + \cot(A) }

\Longrightarrow \rm{ \dfrac{1}{ \sin(A) } + \dfrac{ \cos(A) }{ \sin(A) } }

\Longrightarrow \rm{\dfrac{ 1 + \cos(A) }{ \sin(A) } }

\therefore \boxed{ \small \sf{ \dfrac{ \cot(A) + \csc(A) - 1 }{ \cot(A) - \csc(A) + 1 } = \dfrac{ 1 + \cos(A) }{ \sin(A) } }}

See My Answer for the Same Question On :

https://brainly.in/question/13757817

Answered by Sharad001
90

Question :-

Prove that

 \frac{ \csc a +  \cot a - 1}{1  -  \csc a + \cot a}  =  \frac{1 +  \cos a}{ \sin a}  \\

Used formula :-

 \star \:  { \csc}^{2}  \theta -   { \cot}^{2}  \theta = 1 \\

Proof :-

We will have to show that left hand side (LHS) is equal to right hand side (RHS).

Taking left hand side

 \star \:  \:  \frac{ \csc a +  \cot a - 1}{1  -  \csc a + \cot a}  \:  \\  \\  \star \:  \frac{ \csc a +  \cot a -  \{ { \csc}^{2}a -  { \cot}^{2}a \}  }{1  -  \csc a + \cot a}  \\  \\  \star \:  \frac{ \csc a +  \cot a -  \{ (\csc  a +  \cot a)( \csc a -  \cot a) \}}{1  -  \csc a + \cot a}  \:  \\  \\  \star \:  \frac{ (\csc a +  \cot a) \{1 -  \csc a +  \cot a \}}{1 -  \csc a +  \cot a}  \\  \\  \star \:  \csc a +  \cot a \\  \\  \star \:  \frac{1}{ \sin a}  +  \frac{ \cos a}{ \sin a}  \\  \\  \star \:  \frac{1 +  \cos a}{ \sin a}

left hand side = right hand side

hence proved .

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