91Ó°ÊÓ

We have given that following function :-

$f\left(\mathrm{θ}\right)=sec\mathrm{θ}$and role="math" $f\left(a\right)=-4$.

We have to find the value of $f(-a)$and value of $f\left(a\right)+f(a+\mathrm{Ï€})+f(a+4\mathrm{Ï€})$.

To find value of $f(-a)$we will use even-odd properties and to find the value of $f\left(a\right)+f(a+2\mathrm{Ï€})+f(a+4\mathrm{Ï€})$we will use periodic properties.

We have given that :-

$f\left(\mathrm{θ}\right)=sec\mathrm{θ}$and $f\left(a\right)=-4$.

We know that :-

$sec(-\mathrm{θ})=sec\mathrm{θ}$.

Now put $\mathrm{θ}=-a$, in $f\left(\mathrm{θ}\right)=sec\mathrm{θ}$, then we have :-

$$\begin{gathered} f(-a)=sec(-a) \\ ⇒f(-a)=sec\left(a\right) \\ ⇒f(-a)=f\left(a\right) \end{gathered}$$

Put $f\left(a\right)=-4$, then we have :-

$f(-a)=-4$.

This is the required value.

We have given that :-

$f\left(\mathrm{θ}\right)=sec\mathrm{θ}$.

We know that secant function is of period $2\mathrm{Ï€}$.

This gives us :-

$sec(\mathrm{θ}+2\mathrm{π}k)=sec\mathrm{θ}$, for any integer $k$.

Now :-

localid="1647095654962" $$\begin{gathered} f\left(a\right)+f(a+2\mathrm{π})+f(a+4\mathrm{π})=sec\left(a\right)+sec(a+2\mathrm{π})+sec(a+4\mathrm{π}) \\ ⇒f\left(a\right)+f(a+2\mathrm{π})+f(a+4\mathrm{π})=sec\left(a\right)+sec\left(a\right)+sec\left(a\right) \\ ⇒f\left(a\right)+f(a+2\mathrm{π})+f(a+4\mathrm{π})=3sec\left(a\right) \\ ⇒f\left(a\right)+f(a+2\mathrm{π})+f(a+4\mathrm{π})=3f\left(a\right) \end{gathered}$$

Put $f\left(a\right)=-4$, then we have :-

$$\begin{gathered} f\left(a\right)+f(a+2\mathrm{π})+f(a+4\mathrm{π})=3(-4) \\ ⇒f\left(a\right)+f(a+2\mathrm{π})+f(a+4\mathrm{π})=-12 \end{gathered}$$

This is the required value.