Question

Prove that :

tanA/1-cotA + cotA/1-tanA = secAcosecA+1

Answer 1

tan A/(1-cot A) +cot A/(1-tan A)
=(sin A/cos A) /(1-cos A/sin A) + (cos A/sin A) /(1-sin A/cos A)
=sin²A/cosA(sinA - cosA) +cos²A /sinA (cosA-sinA)
=sin²A/cosA(sinA-cosA) - cos²A/sinA (sinA-cosA)
=(sin³A-cos³A)/sinAcosA(sinA-cosA)
=(sinA-cosA)(sin²A+cos²A+sinAcosA)/sinAcosA(sinA-cosA)
=(1+sinAsinA)/sinAcosA
=(1/sinAcosA)+1
=1+secAcosecA