Question

Prove that cot x /cosec x +1 = cosec x -1/cot x
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Answer 1

Answer:

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Step-by-step explanation:

Answer 2

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

To prove :

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

LHS

$\leadsto \: \frac{ \cot( \alpha ) }{ \csc( \alpha ) + 1}$

Rationalise the denominator,

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

LHS=RHS

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

Formula :

◆csc^2@-cot^2@= 1

◆(a+b)(a-b)= a^2-b^2