It is a fraction and formed by a trigonometric function cosine and angle. In other words, the numerator represents a trigonometric function and the denominator represents angle of the triangle in this fraction.

In Limits, we have a formula that displays a solution for the ratio of a trigonometric function to angle.

The numerator is in sine term but the numerator of the given problem is in cosine term. If the numerator of the problem is transformed into sine term, the formula can be used to solve the problem.

Step: 1

### Transforming the numerator

Let us try to transform the cosine term into sine firstly by the concept of trigonometric ratios of allied angles.

It can be written in $x$ term.

Therefore, the numerator $cosx$ can be replaced by .

Step: 2

### Transforming the denominator

In numerator, sine function contains as angle. If the denominator is transformed into the same form, it will be in the form of the formula which displays a solution for the ratio of sine to angle at angle tends to zero.

is the denominator of the fraction. Let us try to transform it into the angle same as in numerator.

Now, can be replaced by in the denominator of the fraction.

Number $2$ is in denominator and it can be expressed as a multiplying factor of the fractional function.

The rational number $\frac{1}{2}$ is a constant. So, it can be taken out from the function but it should multiply the whole limits function.

Step: 3

### Transforming the Limit

The angle of sine in the numerator is equal to the angle in the denominator. So, the fraction of the problem is in the form of a ratio of sine at angle to angle. So, the formula can be applied but look at the limit of the problem. It is not in the form of the formula.

Therefore, let us transform it.

The limit is given that $x$ tends to $\frac{\pi }{2}$.

It means, if $x$ tends to $\frac{\pi }{2}$, then tends to zero.

Now, the limit can be changed by replacing $x$ tends to $\frac{\pi }{2}$ by the tends to $0$

.

Step: 4

### Getting solution

Consider

We know that

It is the required answer for the given limits problem.

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