Question

prove that (1+tan^A/1+cot^A)=(1-tan A /1-cot A)^=tan^A

Answer 1

hello there!!!!!
we have,

LHS= (1+tan²A/1+cot²A)

=> sec²A/cosec²A [SINCE sec²A=1+tan²A, cosec²A=1+cot²A]

=> 1/cos²A/1/sin²A [since 1/cosA=secA ,1/sinA=cosecA]

=> sin²A/cos²A=tan²A= RHS

middle term= (1-tanA/1-cotA)²

=> [ (1-sinA/cosA)/(1-cosA/sinA)]² [since tanA=sinA/cosA ,cotA=cosA/sinA]

=>(cosA-sinA/cosA)/(sinA-cosA/sinA)]² [taking LCM]

=> [sinA(cosA-sinA)/cosA(sinA-cosA)]²

=> [ -sinA(sinA-cosA)/cosA(sinA-cosA)]²

=> (-sinA/cosA)²= (-tanA)² = tan²A =RHS

Answer 2

Answer is Option D tan²A