Question

prove that [1+1/tan^2A ][1+cot^2A]=1/sin^2A-SIN^4A

Answer 1

The answer won't be equal to RHS
it will be equal if the question is:
(1 + 1/tan²A)(1 + 1/cot²A) = 1/(sin²A - sin^4.A)
LHS
= (1 + 1/tan²A)(1 + 1/cot²A)
= ((tan²A+1)/tan²A)((cot²A+1)/cot²A)
= (sec²A/tan²A)(csc²A/cot²A)
= (sec²A.csc²A)/(tan²A.cot²A)
= sec²A.csc²A
= 1/(sin²A.cos²A)
= (1/sin²A)(1/cos²A)
= (1/sin²A)(1/(1-sin²A))
= 1/(sin²A - sin^4.A)
= RHS