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Question

Prove that sinA/(cotA+cosecA)=2+sinA/(cotA-cosecA)

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Solution

L.H.S = SinA/(cotA+cosecA)

= sinA/(cosA/sinA+1/sinA)

= sinA/{(cosA+1)/sinA}

= sin²A/(1+cosA)

= (1-cos²A)/(1+cosA)

= (1+cosA)(1-cosA)/(1+cosA)

= 1-cosA

L.H.S = 1-cosA

R.H.S = 2+sinA/cotA-cosecA

= 2+sinA/(cosA/sinA-1/sinA)

= 2+sinA/{(cosA-1)/sinA}

= 2+sin²A/(cosA-1)

= 2+(1-cos²A)/{-(1-cosA)}

= 2-(1+cosA)(1-cosA)/(1-cosA)

= 2-(1+cosA)

= 2-1-cosA

= 1-cosA

R.H.S = 1-cosA

∴, LHS=RHS (Proved)


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