Solution We shall prove the result by using principle of mathematical induction We have
P(n) : If 
cos
sin
A
sin
cos
θ
θ


= 

−
θ
θ


, then 
cos
sin
A
sin
cos
n
n
n
n
n
θ
θ


= 

−
θ
θ


, n ∈ N
P(1) : 
cos
sin
A
sin
cos
θ
θ


= 

−
θ
θ


, so 
1
cos
sin
A
sin
cos
θ
θ


= 

−
θ
θ


Therefore,
the result is true for n = 1 Let the result be true for n = k So
P(k) : 
cos
sin
A
sin
cos
θ
θ


= 

−
θ
θ


, then 
cos
sin
A
sin
cos
k
k
k
k
k
θ
θ


= 

−
θ
θ


Now, we prove that the result holds for n = k +1
Now
Ak + 1 =
cos
sin
cos
sin
A A
sin
cos
sin
cos
k
k
k
k
k
θ
θ
θ
θ

 

⋅
= 
 

−
θ
θ
−
θ
θ

 

=
cos cos
– sin sin
cos sin
sin cos
sin cos
cos sin
sin sin
cos cos
k
k
k
k
k
k
k
k
θ
θ
θ
θ
θ
θ +
θ
θ




−
θ
θ +
θ
θ
−
θ
θ +
θ
θ


=
cos(
)
sin(
)
cos(
1)
sin (
1)
sin(
)
cos(
)
sin (
1)
cos(
1)
k
k
k
k
k
k
k
k
θ + θ
θ + θ
+
θ
+
θ




=




−
θ + θ
θ + θ
−
+
θ
+
θ




Therefore, the result is true for n = k  + 1