Question

coseca/coseca-1+coseca/coseca+1=2secsquareA

Answer 1

✅cosecA/cosecA-1+cosecA/cosecA+1

✅=)cosecA(cosecA-1)+cosec(cosecA+1)

,,,,,,,( cosecA-1)(cosecA+1)

✅

=)cosec^2A-cosecA+cosec^2A+cosecA

✅,,,,,,,,,,,cosec^2A-1

✅=)2cosec^2A/cot^2A

✅=)2(1-cot^2A)/cot^2A

=)2tan^2A-2cot^2A/cot^2A

=)2tan^2A-2

RHS proved....

Answer 2

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

$\bullet \; \sf{ \dfrac{ \cosec{A}}{ \cosec{A} - 1} + \dfrac{ \cosec{A}}{ \cosec{A} + 1} = 2 \sec^2{A}}$

⋆ $\bold{\underline{\underline{\rm{\pink{Proof:-}}}}}$

★ Taking L.H.S. :-

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

$\;$

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

$\;$

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

$\;$

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

$\;$

$\implies \sf{ \dfrac{ \dfrac{2}{ \cancel{ \sin^2{A}}}}{ \dfrac{ \cos^2{A}}{ \cancel{ \sin^2{A}}}}}$

$\;$

$\implies \sf{ \dfrac{2}{ \cos^2{A}}}$

$\implies \sf{2 \sec^2{A}}$

✦ LHS = RHS

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

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