Question

$\frac{\mathbf{1}}{\sqrt{\mathbf{c}\mathbf{o}\mathbf{s}\mathbf{\left(}\mathbf{x}\mathbf{\right)}}}$

Short answer

$\left(\cos \right(x){)}^{-1/2}=1+\frac{x^2}{4}+\frac{7x^4}{96}+\frac{139x^6}{5760}+\dots$

Step-by-step solution

Given information.

The Maclaurin expansion of the function $\frac{1}{\sqrt{Cos\left(x\right)}}$is to be evaluated.

Definition of Maclaurin series.

The definition of the Maclaurin series is defined.

$cos\left(x\right)=1-\frac{x^2}{2}+\frac{x^4}{24}-\frac{x^6}{720}+\dots$

$(1+x{)}^{-1/2}=1-\frac{x}{2}+\frac{3x^2}{8}-\frac{5x^3}{16}+\cdots$

Part-A Step 3: Begin by stating the Maclaurin series.

State the Maclaurin series.

$\cos \left(x\right)=1-\frac{x^2}{2}+\frac{x^4}{24}-\frac{x^6}{720}+\dots$

$(1+x{)}^{-1/2}=1-\frac{x}{2}+\frac{3x^2}{8}-\frac{5x^3}{16}+\cdots$

Part-B Step 3: Use the first Maclaurin series to evaluate.

Use the first Maclaurin series and evaluate the expansion.

$\left(\cos \right(x){)}^{-1/2}={\left[1-\frac{x^2}{2}+\frac{x^4}{24}-\frac{x^6}{720}+\cdots \right]}^{-1/2}$

$\left(\cos \right(x){)}^{-1/2}={\left[1+\left[-\frac{x^2}{2}+\frac{x^4}{24}-\frac{x^6}{720}+\dots \right]\right]}^{-1/2}$

Part-C Step 3: Interject by using the second series.

The term $\left[-\frac{x^2}{2}+\frac{x^4}{24}-\frac{x^6}{720}+\dots \right]$is seen as $x$. Evaluate the resulting first Maclaurin Series.

$\left(\cos \right(x){)}^{-1/2}=1-\frac{\left[-\frac{x^2}{2}+\frac{x^4}{24}-\frac{x^6}{720}+\dots \right]}{2}+\frac{3{\left[-\frac{x^2}{2}+\frac{x^4}{24}-\frac{x^6}{720}+\dots \right]}^2}{8}-\frac{5{\left[-\frac{x^2}{2}+\frac{x^4}{24}-\frac{x^6}{720}+\dots \right]}^3}{16}+\dots$

$=1+\frac{276480x^2+80640x^4+26688x^6}{1105920}+\dots$

$\left(\cos \right(x){)}^{-1/2}=1+\frac{x^2}{4}+\frac{7x^4}{96}+\frac{139x^6}{5760}+\dots$

Thus, the answer is $\left(\cos \right(x){)}^{-1/2}=1+\frac{x^2}{4}+\frac{7x^4}{96}+\frac{139x^6}{5760}+\dots$