Question

prove that cot A + cosec a minus one upon cot a minus cosec A + 1 is equal to 1 + Cos A upon sin a​

Answer 1

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

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

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

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

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

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

⠀

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

Answer 2

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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