### Video Transcript

Determine the variation function of π of π₯ is equal to the cos of π₯ at π₯ is equal to π by two.

In this question, weβre asked to determine the variation function of a given trigonometric function: π of π₯ is the cos of π₯. And weβre told to do this at a value of π₯ is equal to π by two. And to do this, letβs start by recalling what we mean by the variation function of a given function as a given value.

We can recall that the variation function gives us a measure of how a function changes when its π₯-inputs change from π to π plus β. In particular, the variation function π£ of β of a function π of π₯ at π₯ is equal to β is given by π evaluated at π plus β minus π evaluated at π. So, in our case, our function π of π₯ is the cos of π₯ and our value of π is π by two.

So, if we substitute our function π of π₯ is equal to the cos of π₯ and π is equal to π by two into this equation, we get π£ of β is equal to the cos of π by two plus β minus the cos of π by two. This is one expression for the variation function π£ of β.

However, we can simplify this expression. First, we know the cos of π by two is equal to zero. This means π£ of β is equal to the cos of π by two plus β. We could leave our answer like this. However, we can simplify this even further. We do this by recalling one of the cofunction identities. The cos of π by two plus π is equal to negative the sin of π. Therefore, the cos of π by two plus β is just equal to negative the sin of β. And we canβt simplify this any further.

So we were able to show the variation function of the function π of π₯ is equal to the cos of π₯ at π₯ is equal to π by two is π£ of β is equal to negative the sin of β.