91Ó°ÊÓ

For the solution use a standard trigonometric identity with $C_1=Acos\mathrm{Ï•},C_2=Asin\mathrm{Ï•}$.

In $C_{1}cos\mathrm{Ó\neg }t+C_{2}sin\mathrm{Ó\neg }t$, we will use $C_1=Acos\mathrm{Ï•},C_2=Asin\mathrm{Ï•}$

Therefore,

$$\begin{gathered} C_{1}cos\mathrm{Ó\neg }t+C_{2}sin\mathrm{Ó\neg }t=Acos\mathrm{Ï•}cos\mathrm{Ó\neg }t+Asin\mathrm{Ï•}sin\mathrm{Ó\neg }t \\ =A\left(cos\mathrm{Ï•}cos\mathrm{Ó\neg }t+sin\mathrm{Ï•}sin\mathrm{Ó\neg }t\right) \\ =Acos\left(\mathrm{Ó\neg }t-\mathrm{Ï•}\right)(U\sin gidentitycos\left(A-B\right)=cosAcosB+sinAsinB) \end{gathered}$$

Hence, role="math" localid="1664172041931" ${\mathit{C}}_{\mathbf{1}}\mathit{c}\mathit{o}\mathit{s}\mathit{Ó\neg }\mathit{t}\mathbf{+}{\mathit{C}}_{\mathbf{2}}\mathit{s}\mathit{i}\mathit{n}\mathit{Ó\neg }\mathit{t}$can be written in the form role="math" localid="1664172054217" $\mathit{A}\mathit{c}\mathit{o}\mathit{s}\left(\mathrm{Ó\neg }t-\mathrm{Ï•}\right)$.

We have, $F\left(\mathrm{t}\right)={[1+(\mathrm{Ó\neg }/\mathrm{k}\left)\right]}^{-\frac{1}{2}}cos(\mathrm{Ó\neg t}-\mathrm{Ï•})$

Comparing its R.H.S. with $Acos\left(\mathrm{Ó\neg }t-\mathrm{Ï•}\right)$,

$$\begin{gathered} A={[1+(\mathrm{Ó\neg }/\mathrm{k}\left)\right]}^{-\frac{1}{2}} \\ A=\frac{K}{\sqrt{K^2+{\mathrm{Ó\neg }}^2}} \end{gathered}$$

Now as given $C_1=Acos\mathrm{Ï•},C_2=Asin\mathrm{Ï•}$

Therefore,

$C_1=\frac{K}{\sqrt{K^2+{\mathrm{Ó\neg }}^2}}cos\mathrm{Ï•},C_2=\frac{K}{\sqrt{K^2+{\mathrm{Ó\neg }}^2}}sin\mathrm{Ï•}$

Substituting these values of C₁ and C₂ in $A=\sqrt{C_1^2+C_2^2}$,

$$\begin{gathered} A=\sqrt{\frac{K^2}{K^2+{\mathrm{Ó\neg }}^2}co{s}^2\mathrm{Ï•}+\frac{K^2}{K^2+{\mathrm{Ó\neg }}^2}si{n}^2\mathrm{Ï•}} \\ A=\sqrt{\frac{K^2}{K^2+{\mathrm{Ó\neg }}^2}\left(co{s}^2\mathrm{Ï•}+si{n}^2\mathrm{Ï•}\right)} \\ A=\sqrt{\frac{K^2}{K^2+{\mathrm{Ó\neg }}^2}} \\ A=\frac{K}{\sqrt{K^2+{\mathrm{Ó\neg }}^2}} \\ A=\frac{1}{\sqrt{1+{(\mathrm{Ó\neg }/k)}^2}} \end{gathered}$$

Hence, the representation role="math" localid="1664173344588" $\mathit{F}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}{\mathbf{[}\mathbf{1}\mathbf{+}\mathbf{(}\mathbf{Ó\neg }\mathbf{/}\mathbf{k}\mathbf{\left)}\mathbf{\right]}}^{\mathbf{-}\frac{\mathbf{1}}{\mathbf{2}}}\mathit{c}\mathit{o}\mathit{s}\mathbf{(}\mathbf{Ó\neg t}\mathbf{-}\mathbf{Ï•}\mathbf{)}$is verified according to the given conditions .