91Ó°ÊÓ

We have given that following function :-

$f\left(\mathrm{θ}\right)=cot\mathrm{θ}$and $f\left(a\right)=-3$.

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 $f\left(a\right)+f(a+\mathrm{Ï€})+f(a+4\mathrm{Ï€})$we will use periodic properties.

We have given that :-

$f\left(\mathrm{θ}\right)=cot\mathrm{θ}$and $f\left(a\right)=-3$.

We know that :-

$cot(-\mathrm{θ})=-cot\mathrm{θ}$.

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

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

Put $f\left(a\right)=-3$, then we have :-

$$\begin{gathered} f(-a)=-(-3) \\ ⇒f(-a)=3 \end{gathered}$$

This is the required value.

We have :-

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

We know that cotangent function is periodic of period $\mathrm{Ï€}$.

This gives us :-

$cot(\mathrm{θ}+\mathrm{π}k)=cot\mathrm{θ}$.

Now :-

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

Put $f\left(a\right)=-3$, then we have :-

$$\begin{gathered} f\left(a\right)+f(a+\mathrm{π})+f(a+4\mathrm{π})=3(-3) \\ ⇒f\left(a\right)+f(a+\mathrm{π})+f(a+4\mathrm{π})=-9 \end{gathered}$$

This is the required value.