Question

tanA/1-cotA+cotA/1-tanA=sinAcosecA+1

Answer 1

Hey dear! I guess on the RHS it should be 'sec A Cosec A +1.'

Ques :- (tan A) / (1 - cot A) + (cot A) /( 1 - tan A) = sec A cosec A + 1

Solution:

LHS - (tan A) / (1 - cot A) + (cot A) /( 1 - tan A)

= (tan A) / ( 1 - 1/tan A) + ( 1/ tan A) / (1 - tan A)

= tan²A / (tan A - 1) + 1 / ( tan A - tan²A)

= - tan ²A / (1- tan A) + 1 / tan A( 1 - tan A)

= (- tan³A + 1) / tan A (1 - tan A)

= (1³ - tan³A ) / tan A (1 - tan A)

[Now apply the identity .......a³ - b³ = (a - b) (a² + b² + ab)]

= {( 1 - tan A) (1 + tan²A + tan A)} / tan A ( 1 - tan A)

= (1 - tan²A + tan A) / tan A

= (sec²A + tan A) / tan A

= sec²A / tan A + tan A/tan A

= sec²A cot A + 1

= ( 1/ cos ²A × cos A/ sin A) + 1

= 1 / cos A × 1 / sin A + 1

= sec A cosec A + 1 <= RHS

Hence proved!!

Hope it helps!!^^