Solve each equation , then write the general solution

cos(pi/2(x+1))=cospi/2 x

x=1.5, 3.5
general solution: 1/2+4n
but the back of the book says that the general solution is 1/2+2n

isn't the period 4?

cos(pi/2(x+1))=cospi/2 x

cos((pi/2)x + pi/2) = cos(pi/2)x
cos((pi/2)x)cos(pi/2) - sin((pi/2)x)sin(pi/2) = cos (pi/2)x
0 - sin((pi/2)x = cos (pi/2)x
divide both sides by -cos(pi/2)x
tan (pi/2)x = -1

we know tan (pi/2) = +1 , so (pi/2)x must in the II or IV quadrant.

so (pi/2)x = pi - pi/2 or (pi/2)x = 2pi - pi/2

x = 1/2 or x = 3/2

but the period of tan (pi/2)x = 2
(unlike the sine or cosine which would indeed by 4, but I was solving tan at the end)

so the general solution is
1/2 + 2n but also 3/2 + 2n

the book missed that one, but when you sub in 3/2 it works in the original equation, and the general solution of
1/2 + 2n does not produce 3/2

hmm.. how did you get

cos((pi/2)x)cos(pi/2) - sin((pi/2)x)sin(pi/2) = cos (pi/2)x

also, do you have to solve for tan?

I just learnt to find the intersections of these 2 equations on a graphing calc. and then write the general solution based on the intersections.

To solve the equation cos(pi/2(x+1)) = cos(pi/2x), we can start by equating the arguments of the cosine functions:

pi/2(x+1) = pi/2x

Now, let's solve for x:

pi/2x + pi/2 = pi/2x

pi/2 = 0

This equation leads to a contradiction since pi/2 is not equal to 0. Therefore, there are no solutions to this equation.

However, if we made a mistake in our calculations or if there is a typo in the given equation, we can certainly check for possible solutions.

Assuming the equation is cos(pi/2(x+1)) = cos(pi/2x), let's find the values of x that satisfy this equation:

cos(pi/2(x+1)) = cos(pi/2x

Since the cosine function is equal when the angles are coterminal, we can write:

pi/2(x+1) = 2n*pi ± pi/2x, where n is an integer

Expanding and rearranging the equation, we get:

x = 2n + 1/2 ± 1/2

This simplifies to:

x = 2n + 1 ± 1/2

So, the possible solutions for x are:

x = 2n + 1/2 + 1/2 = 2n + 1

Therefore, the general solution for the equation cos(pi/2(x+1)) = cos(pi/2x is x = 2n + 1, where n is an integer.

Based on the information provided, it seems like the book may have made an error in stating the general solution as 1/2 + 2n. The correct general solution should be x = 2n + 1.