State true or false.If 3 cot A = 4, then

1−

tan
2

A

1+

tan
2

A

=

cos
2

A−

sin
2

A
.

Question

State true or false.If 3 cot A = 4, then

1−
      
       tan
      2
     
     A
    
     1+
      
       tan
      2
     
     A
   
   =
    
     cos
    2
   
   A−
    
     sin
    2
   
   A
  .

Accepted Answer

Given that 3 cot A = 4.Then,

cotA=
    4
    3

.Now, cotangent is a ratio of the side adjacent to angle A by side opposite to angle A.

cotA=
      4
      3

⇒
      
       Adjacent side
      
       Opposite side
     
     =
      
       4k
      
       3k

where k is the positive integer.Form a triangle with respect to A with opposite side as BC and adjacent side as AB.By applying Pythagoras theorem, we get,

A
      C
      2
     
     =B
      C
      2
     
     +A
      B
      2

⇒A
      C
      2
     
     =9
      k
      2
     
     +16
      k
      2

⇒A
      C
      2
     
     =25
      k
      2

⇒AC=5k

Now,

sinA=
      
       Opposite side
      
       Hypotenuse

⇒sinA=
      
       BC
      
       AC

⇒sinA=
      
       3k
      
       5k

⇒sinA=
      3
      5

cosA=
      
       Adjacent side
      
       Hypotenuse

⇒cosA=
      
       AB
      
       AC

⇒cosA=
      
       4k
      
       5k

⇒cosA=
      4
      5

tanA=
      
       Opposite side
      
       Adjacent side

⇒tanA=
      
       BC
      
       AB

⇒tanA=
      
       3k
      
       4k

⇒tanA=
      3
      4

Then,

1−
      
       tan
      2
     
     A
    
     1+
      
       tan
      2
     
     A
   
   =
    
     cos
    2
   
   A−
    
     sin
    2
   
   A
  .Substitute the values in the above expression, we get,

⇒
    
     1−

3
          4

2

1+

3
          4

2

=

4
        5

2
   
   −

3
        5

2

⇒
    
     1−
      9
      
       16

1+
      9
      
       16

=
    
     16
    
     25
   
   −
    9
    
     25

⇒
    
     16−9
    
     16
   
   ×
    
     16
    
     16+9
   
   =
    
     16−9
    
     25

⇒
    7
    
     25
   
   =
    7
    
     25

Thus, the given expression is true.Hence, option 1 is correct.

State true or false.

If 3 cot A = 4, then $1 − tan 2 A 1 + tan 2 A = cos 2 A − sin 2 A$ .

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State true or false.If 3 cot A = 4, then 1− tan 2 A 1+ tan 2 A = cos 2 A− sin 2 A .