Question

prove that tan∅/1-cot∅+cot∅/1-tan∅=1+sec∅.cosec∅​

Answer 1

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

= (tan A)/[(1 - (1/tan A)] + cot A /(1 - tan A)

= (tan2 A)/[(tan A - 1)] + cot A /(1 - tan A)

= (tan2 A)/[(tan A - 1)] - cot A /(tan A - 1)

= (tan2 A - cot A) / (tan A - 1)

= (tan2 A - 1/tan A) / (tan A - 1)

= (tan3 A - 1) / [tan A (tan A - 1)]

= (tan A - 1)(tan2 A + tan A + 1) / [tan A (tan A - 1)]

= (tan2 A + tan A + 1) / tan A

= 1 + tan A + cot A

= 1 + [(sin A/cosA) + (cos A/sin A)]

= 1 + [(sin2 A + cos2 A) / sin A cos A]

= 1 + [1 / (sin A cos A)]

= 1 + (sec A x cos A)

Answer 2

Answer:

sinx/cosx*1-cosx/sinx+cosx/sinx*1-sinx/cosx

sinx/cosx*sinx-cosx/sinx+cosx/sinx*cosx-sinx/cosx

1+secx.cosecx