4
Rationalised 2023-24
 70
MATHEMATICS
Miscellaneous Examples
Example 23 If 
cos
sin
A
sin
cos
θ
θ


= 

−
θ
θ


, then prove that 
cos
sin
A
sin
cos
n
n
n
n
n
θ
θ


= 

−
θ
θ


, n ∈ N 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