Question

 p.t (1+tana+seca) (1+cota-coseca)=2

Answer 1

(1 + tan a + sec a )(1+ cot a - cosec a)

= 1 + cot a - cosec a  + tan a + 1 - sec a + sec a + cosec a - sec a.cosec a
= 2 + cot a + tan a - sec a . cosec a
= 2 + (cos a)/(sin a) + (sin a)/(cos a) - sec a . cosec a
= 2 + (cos² a + sin² a)/(sin a. cos a) - sec a . cosec a
= 2 + sec a . cosec a - sec a . cosec a
= 2 proved

Answer 2

(1 + tan a + sec a )(1+ cot a - cosec a)

1 + cot a - cosec a  + tan a + tan a.cot a - tan a.cosec a + sec a + sec a.cot a - sec a.cosec a

[ cosec a get cancelled by - cosec a ]
[tan a = 1/tan a and get cancelled and become 1]
[tan a = sin a / cos a and its reciprocal become sec a / cosec a ]
[cot a = cos a / sin a and its reciprocal becomes cosec a / seca]

1 + cot a - cosec a  + tan a + 1 - sec a + sec a + cosec a - sec a.cosec a

2 + (cos a)/(sin a) + (sin a)/(cos a) - sec a . cosec a

2 + (cos² a + sin² a)/(sin a. cos a) - sec a . cosec a

 2 + sec a . cosec a - sec a . cosec a

2.

Hence proved