Is the following true or false?If

sin

90
°

−θ
+cos

90
°

−θ
=
2

sin

90
°

−θ

then

(sinθ−cosθ)=
2

sinθ
.

Question

Is the following true or false?If

sin

90
      °
     
     −θ
   +cos

90
      °
     
     −θ
   =
    2
   
   sin

90
      °
     
     −θ
   
   then

(sinθ−cosθ)=
    2
   
   sinθ
  .

Accepted Answer

Given that

sin

90
      °
     
     −θ
   +cos

90
      °
     
     −θ
   =
    2
   
   sin

90
      °
     
     −θ
   
  .We use,

sin

90
        °
       
       −θ
     =cosθ;

cos

90
        °
       
       −θ
     =sinθ;

a+b
      
      2
     
     =
      a
      2
     
     +
      b
      2
     
     +2ab;

a−b
      
      2
     
     =
      a
      2
     
     +
      b
      2
     
     −2ab

Now,

sin

90
      °
     
     −θ
   +cos

90
      °
     
     −θ
   =
    2
   
   sin

90
      °
     
     −θ

cosθ+sinθ=
    2
   
   cosθ
    On squaring we get,

cosθ+sinθ
     
    2
   
   =2
    
     cos
    2
   
   θ

cos
    2
   
   θ+
    
     sin
    2
   
   θ+2cosθsinθ=2
    
     cos
    2
   
   θ

sin
    2
   
   θ=2
    
     cos
    2
   
   θ−
    
     cos
    2
   
   θ−2cosθsinθ
   Adding

sin
    2
   
   θ
   on both sides,

sin
    2
   
   θ+
    
     sin
    2
   
   θ=
    
     cos
    2
   
   θ+
    
     sin
    2
   
   θ−2cosθsinθ

2
    
     sin
    2
   
   θ=

sinθ−cosθ
     
    2

sinθ−cosθ=
    2
   
   sinθ
   Thus the given expression is true.Therefore the correct option is 1.

Is the following true or false?

If $sin 90 ° − θ + cos 90 ° − θ = 2 sin 90 ° − θ$ then $(sin θ − cos θ) = 2 sin θ$ .