Question

This was that question plzz hrlp i cant understand

Answer 1

hope it helps you
sorry it took much time

Answer 2

sinΘ - cosΘ + 1 
_____________     =  1 /secΘ - tanΘ

sinΘ + cosΘ - 1  


Taking LHS

     sinΘ - cosΘ + 1 
=   _____________

    (sinΘ + cosΘ) - 1

   sinΘ - cosΘ + 1              (sinΘ + cosΘ) + 1 
= _____________  ×       ________________

  (sinΘ + cosΘ) - 1             (sinΘ + cosΘ) + 1 
 
(Taking Conjugate)

   (sinΘ - cosΘ + 1)(sinΘ + cosΘ + 1)  
= _____________________________

   (sinΘ + cosΘ - 1)(sinΘ + cosΘ + 1)

    sin²Θ + sinΘcosΘ + sinΘ - sinΘcosΘ - cos²Θ - cosΘ + sinΘ + cosΘ + 1
= ________________________________________________________

    sin²Θ + sinΘcosΘ + sinΘ + sinΘcosΘ + cos²Θ + cosΘ - sinΘ - cosΘ - 1

Cancelling out,

       sin²Θ + 2sinΘ + (1 - cos²Θ)
=   __________________________

   (sin²Θ + cos²Θ) - 1 + 2sinΘcosΘ

    sin²Θ + 2sinΘ + sin²Θ
=   ___________________                                       (By sin²Θ + cos²Θ =1)

      1 - 1 + 2sinΘcosΘ

    2sin²Θ + 2sinΘ
=  _____________

      2sinΘcosΘ
   
    2sinΘ(sinΘ+1)
= _____________

       2sinΘ(cosΘ)

    sinΘ + 1
= ________

      cosΘ

= tanΘ + secΘ                      ...............(1)


Taking RHS

         1
= ________

   tanΘ - secΘ

          1              tanΘ + secΘ
= ________ ×   __________

   tanΘ - secΘ    tanΘ + secΘ

(Taking Conjugate)

        tanΘ + secΘ
=   _____________

       tan²Θ - sec²Θ

= tanΘ + secΘ                 ................(2)             (By sec²Θ = 1 + tan²Θ)

From (1) and (2)

We get,   LHS = RHS


HENCE PROVED

☺ Hope this Helps ☺