show that cos(pi x) = pi cosx is not an identity by finding a single value of x for which it fails to hold.
How about the most obvious case, x - 0
LS = cos (π(0)) = cos 0 = 1
RS = 0(cos0) = 0(1) = 0 ≠ LS
Now, a more interesting question would have been ,
For what values of x is the statement true?
http://www.wolframalpha.com/input/?i=cos%28πx%29+%3D+π*cos%28x%29
To show that cos(pi x) = pi cos(x) is not an identity, we need to find a single value of x for which the equation does not hold.
Let's assume the equation is an identity. If it were an identity, it would have to hold for all possible values of x. So, we can substitute a specific value for x and see if the equation holds true.
Let's substitute x = 1 into the equation:
cos(pi x) = pi cos(x)
cos(pi) = pi cos(1)
Using the value of pi (approximately 3.14159) and calculating cos(1) using a calculator, we get:
-1 = 3.14159 * cos(1)
Upon evaluating the right-hand side of the equation, we find that -1 does not equal 3.14159 * cos(1). Therefore, the equation cos(pi x) = pi cos(x) is not an identity.
Thus, we have shown that the equation is not universally true for all values of x by providing a counterexample when x = 1.