The given function is analytic at a = 0. Use appropriate series in (2) and long division to find the first four nonzero terms of the Maclaurin series of the given function. Sec x.

Step-by-step explanation:

From the given information:

we are to find the first four nonzero terms of the Maclaurin series of the given function.

Sec x.

If we recall ; we will realize that the derivative of sec x =

Also; for cos x ; the first four terms of its Maclaurin Series can be expressed as ;

=

However, using the long division method: we have;

the rule of the long division method is to first use the 1 from the denominator to divide the 1 from the numerator. the multiply it with the answer we get which is (1) before subtracting it from  that answer (1).

i.e

1/1 = 1

1 × 1 = 1

1 - 1 = 0

Afterwards; we will subtract the remaining integers from this numerator.

So, we have:

We are going to apply the same process to the remainder ;

which is to divide the second integer with 1

Then we will multiply the numerator with ; the result will then be subtracted from the polynomial.

Repeating the same process for remainder  ; we have:

so; we will need to multiply 1 with and subtract it from the rest of the polynomial

Here ; the final remainder is ; repeating the usual process for long division method; we have:

So;

Now the  first four nonzero terms of the Maclaurin series is the addition of all the integers used as remainders ; i.e

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step-by-step explanation:

i think the answer would be c