Question

Prove that 1+ cot 2 A/1+cosecA = 1/sinA

Answer 1

1+(cot²A/1+cosecA)
=1+[(cos²A/sin²A)/(1+1/sinA)]
=1+[(cos²A/sin²A)/{(sinA+1)/sinA}]
=1+[cos²A/sin²A×sinA/(sinA+1)]
=1+[cos²A/sinA(sinA+1)]
=[sinA(1+sinA)+cos²A]/sinA(sinA+1)
=(sinA+sin²A+cos²A)/sinA(sinA+1)
=(sinA+1)/sinA(sinA+1)
=1/sinA (Proved)