Help me guys......
Prove that :-tanA/(1-cotA) + cotA/(1-tanA) = secA*cosecA+1.
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LHS = tanA/1-cotA + cotA/1-tanA
= sinA/cosA/1-cosA/sinA + cosA/sinA/1-sinA/cosA
= sinA/cosA/sinA-cosA/sinA + cosA/sinA/cosA-sinA/cosA
= sin²A/cosA(sinA-cosA) + cos²A/(cosA-sinA)sinA
= sin²A/cosA(sinA-cosA) - cos²A/(sinA-cosA)
= sin³A-cos³A/(sinA-cosA)(sinAcosA)
= (sinA-cosA)(sin²A+cos²A + sinAcosA)/(sinA-cosA)(sinAcosA) [By using a³-b³ = (a-b) (a²+ab+b²)]
= 1+sinAcosA/sinAcosA
= 1/sinAcosA + sinAcosA/sinAcosA
= secA*cosecA + 1 = RHS
= sinA/cosA/1-cosA/sinA + cosA/sinA/1-sinA/cosA
= sinA/cosA/sinA-cosA/sinA + cosA/sinA/cosA-sinA/cosA
= sin²A/cosA(sinA-cosA) + cos²A/(cosA-sinA)sinA
= sin²A/cosA(sinA-cosA) - cos²A/(sinA-cosA)
= sin³A-cos³A/(sinA-cosA)(sinAcosA)
= (sinA-cosA)(sin²A+cos²A + sinAcosA)/(sinA-cosA)(sinAcosA) [By using a³-b³ = (a-b) (a²+ab+b²)]
= 1+sinAcosA/sinAcosA
= 1/sinAcosA + sinAcosA/sinAcosA
= secA*cosecA + 1 = RHS
Anonymous:
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