Question

cotA+cosecA-1/cotA-cosec+1= 1+cosA/sinA​

Answer 1

⋆ $\bold{\underline{\underline{\rm{\red{To \; Prove:}}}}}$

$\bullet \sf{ \dfrac{ \cot{A} + \cosec{A} - 1}{ \cot{A} - \cosec{A} + 1} = \dfrac{1 + \cot{A}}{ \sin{A}}}$

⋆ $\bold{\underline{\underline{\rm{\blue{Proof:}}}}}$

★ Taking L.H.S:-

$\implies \sf{ \dfrac{ \cosec{A} + \cot{A} - 1}{ \cot{A} - \cosec{A} + 1}}$

★ Identity used:-

$\bold{\underline{\underline{\boxed{\sf{\purple{\dag \; \cosec^2{A} - \cot^2{A} = 1}}}}}}$

★ Substituting this in the numerator:-

$\implies \sf{ \dfrac{ \cosec{A} + \cot{A} - ( \cosec^2{A} - \cot^2{A})}{ \cot{A} - \cosec{A} + 1}}$

✦ We know that,

$\implies \sf{x^2 + y^2 = (x + y)(x - y)}$

$\implies \sf{ \dfrac{ \cosec{A} + \cot{A} - ( \cosec{A} + \cot{A})( \cosec{A} - \cot{A})}{ \cot{A} - \cosec{A} + 1}}$

$\implies \sf{ \dfrac{( \cosec{A} + \cot{A})( 1 - \cosec{A} + \cot{A})}{ \cot{A} - \cosec{A} + 1}}$

$\implies \sf{ \dfrac{( \cosec{A} + \cot{A}) \cancel{( 1 - \cosec{A} + \cot{A})}}{ \cancel{( \cot{A} - \cosec{A} + 1)}}}$

$\implies \sf{ \cosec{A} + \cot{A}}$

$\implies \sf{ \dfrac{1}{sin{A}} + \dfrac{ \cos{A}}{ \sin{A}}}$

$\implies \sf{ \dfrac{1 + \cos{A}}{ \sin{A}}}$

$\bold{\underline{\underline{\sf{\purple{\dag \; Hence \; Proved!}}}}}$

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