hello my name is Chris, student
 
I need some help of how to solve the problem

"use the principle of mathematical induction to prove that the following are true for all positive integers"

cos(n x pi + X) = (-1)^n cosX

any help would be appreciated
 
 

Hi Chris,

There are two steps in an induction argument. First you need to get started, that is you need to verify that the statement is true when n=1.

n=1:

The statement is then cos(1 x pi + X) = (-1)¹ cosX or
cos(pi + X) = -1 cosX

I would use an identity to verify this statement. The identity is

cos(A + B) = cos(A)cos(B) - sin(A)sin(B)

Hence cos(pi + X) = cos(pi)cos(X) - sin(pi)sin(X).
But cos(pi) = -1 ans sin(pi) = 0 so

cos(pi + X) = -1 cos(X) - 0 = -1 cos(X) as required.

The second step is to assume that the statement is true for some value of n and then show it is true for the next value of n. I will assume that the statement is true when n=k.

Assume cos(k x pi + X) = (-1)^k cosX:

Verify that the statement is true for the next value of k. That is show that

cos([k+1] x pi + X) = (-1)^k+1 cosX:

Look at the left side and rewrite it as

cos([k+1] x pi + X) = cos(k x pi + pi + x) = cos(pi + k x pi + x)
= cos(pi + [k x pi + X])

This is again of the form cos(A + B) so I can use the identity from above.

cos(pi + [k x pi + X])
= cos(pi)cos( [k x pi + X]) - sin(pi)sin([k x pi + X])
= -1 cos([k x pi + X]) - 0
= -1 cos([k x pi + X])

But we are working under the assumption that cos(k x pi + X) = (-1)^k cosX so

-1 cos([k x pi + X]) = (-1) (-1)^k cosX = (-1)^k+1cosX

Thus the statement is true when n=1, and if it is true for any value of n it is true for the next value of n. Thus the statement is true for all positive integers.

Penny

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