RJC year 2000 Maths S Paper question :D
Very fun to do one. Do try! It tests very much on the concepts of induction.
Show by induction that, for every positive integer n, both
cos (nx) and sin (nx) / sin x
can be expressed as polynomials in cos x with integer coefficients.
Answer:
Let Pn be the statement: cos (nx) can be expressed polynomials in cos x with integer coefficients
Let Gn be the statement: sin (nx) / sin x can be expressed polynomials in cos x with integer coefficients
For cos (nx),
n=1 => cos x
n=2 => 2 cos2 x - 1
P1 and P2 are true
For sin (nx) / sin x,
n=1 => 1
n=2 => 2 cos x
G1 and G2 are true
Suppose Pk and Gk are true.
Pk+1:
cos ([k+1]x) = cos kx cos x - sin kx sin x
= cos kx cos x - (sin kx / sin x) (sin2 x)
= cos kx cos x - (sin kx / sin x) (1 - cos2 x)
Since cos kx can be expressed as a polynomial in cos x with integer coefficients, cos kx cos x can also be expressed as a polynomial in cos x with integer coefficients.
Since sin kx / sin x can be expressed as a polynomial in cos x with integer coefficients, (sin kx / sin x) (1 - cos2 x) can also be expressed as a polynomial in cos x with integer coefficients.
Therefore, suppose Pk and Gk are true, Pk+1 is true.
Gk+1:
sin ([k+1]x) / sin x = (sin kx cos x + cos kx sin x) / sin x
= (sin kx / sin x) cos x + cos kx
cos kx can be expressed as a polynomial in cos x with integer coefficients
Since sin kx / sin x can be expressed as a polynomial in cos x with integer coefficients, (sin kx / sin x) (cos x) can also be expressed as a polynomial in cos x with integer coefficients.
Therefore, suppose Pk and Gk are true, Gk+1 is true.
Combining, Pk and Gk are true implies that Pk+1 and Gk+1 is true
But P1, P2, G1 and G2 are true, i.e. P1 and G1 true implies that P2 and G2 are true. P2 and G2 true implies that P3 and G3 are true. And so on...
Hence, by mathematical induction, Pn and Gn are true for every positive integer n.
i.e. Both cos (nx) and sin (nx) / sin x can be expressed as polynomials in cos x with integer coefficients.
