Question

(CosecA-cotA)square=1-cosA/1+cosa

Answer 1

AnswEr :

• To Prove :

$\rm( \csc(A) - \cot(A) )^{2} = \dfrac{1 - \cos(A) }{1 + \cos(A) }$

• Proof :

I'm taking RHS to Prove this Identity -

$\longrightarrow \sf\dfrac{1 - \cos(A) }{1 + \cos(A) }$

$\longrightarrow \sf\dfrac{1 - \cos(A) }{1 + \cos(A) } \times 1$

⠀⠀⠀⋆ Multiplying by 1 - cos(A) / 1 - cos(A)

$\longrightarrow \sf\dfrac{1 - \cos(A) }{1 + \cos(A) } \times \dfrac{1 - \cos(A) }{1 - \cos(A) }$

$\longrightarrow \sf\dfrac{(1 - \cos(A) )^{2} }{1 - \cos^{2} (A) }$

⠀⠀⠀⋆ (a - b)² = a² + b² - 2ab

⠀⠀⠀⋆ (1 - cos²(A)) = sin²(A)

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

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

⠀⠀⠀⋆ 1 / sin²(A) = cosec²(A)

⠀⠀⠀⋆ cos²(A) / sin²(A) = cot²(A)

⠀⠀⠀⋆ cos(A) / sin²(A) = cot(A)cosec(A)

$\longrightarrow \sf \csc^{2} (A) + \cot^{2} (A) - 2 \cot(A) \csc(A)$

⠀⠀⠀⋆ (a² + b² + 2ab) = (a - b)²

$\longrightarrow \large \sf( \csc(A) - \cot(A) )^{2}$

⠀

$\therefore \boxed{ \rm( \csc(A) - \cot(A) )^{2} = \dfrac{1 - \cos(A) }{1 + \cos(A) } }$

Answer 2

$\huge \red{ \mathfrak{solution}}$

To prove :

${( \csc( \alpha ) - \cot( \alpha )) }^{2} = \frac{1 - \cos( \alpha ) }{1 + \cos( \alpha ) }$

LHS

$\leadsto \: ( { \csc( \alpha ) - \cot( \alpha )) }^{2} \\ \\ \leadsto \:( \frac{1}{ \sin( \alpha ) } - \frac{ \cos( \alpha ) }{ \sin( \alpha ) } ) ^{2} \\ \\ \leadsto( \frac{1 - \cos( \alpha ) }{ \sin( \alpha ) } )^{2} \\ \\ \leadsto \: \frac{(1 - \cos( \alpha )) ^{2} }{ { \sin }^{2} \alpha } \\ \\ \leadsto \: \frac{( {1 - \cos( \alpha ) })^{2} }{1 - \cos ^{2} ( \alpha ) } \\ \\ \leadsto \: \frac{(1 - \cos( \alpha ) )(1 - \cos( \alpha ) )}{(1 - \cos( \alpha ))(1 + \cos( \alpha )) } \\ \\ \leadsto \: \cancel \frac{1 - \cos( \alpha ) }{1 - \cos( \alpha ) } \times \frac{1 - \cos( \alpha ) }{1 + \cos( \alpha ) } \\ \\ \leadsto \: \frac{1 - \cos( \alpha ) }{1 + \cos( \alpha ) }$

LHS = RHS

$\boxed{ \green{\bold{ \mathfrak{proved}}}}$

Formula :

• a^2-b^2=(a+b)(a-b)

• csc@= 1/sin@

• cot@= cos@/sin@