Question

29th question
25 points

Answer 1

Hi,

LHS = sinA-cosA+1/sinA+cosA-1

divide both numerator and denominator by cosA

LHS=(tanA−1+secA)/(tanA+1−secA)LHS=(tanA−1+secA)/(tanA+1−secA)

Now

sec2A=1+tan2Asec2A=1+tan2A

sec2A−tan2A=1sec2A−tan2A=1

Using above relation at denominator of LHS

LHS=(tanA−1+secA)/(tanA−secA+sec2A−tan2A)LHS=(tanA−1+secA)/(tanA−secA+sec2A−tan2A)

LHS=(tanA−1+secA)/((secA−tanA)(−1+secA+tanA))LHS=(tanA−1+secA)/((secA−tanA)(−1+secA+tanA))

LHS=1/(secA−tanA)LHS=1/(secA−tanA)

LHS=RHSLHS=RHS

Hence Proved.

I think above proof will clear your doubt,

All the best.

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 ☺