What will be the value of $\cos (n\pi x /l)$ for $x=0$?

Question

$\cos0 = 1$ but the value of the above expression is $1$. How it is equal to $1$? If $\cos0 =1$ then the value should be $n\pi/l$.

Answer 1

If the quantity were $\frac{cos(x)n\pi}{l}$ then the value would be $\frac{(1)n\pi}{l}= \frac{n\pi}{l}$. But it is not! You have parentheses around all of $n\pi x$ so the cosine function is applied to $n\pi x= n\pi 0= 0$. The correct value is $\frac{cos(n\pi 0)}{l}= \frac{cos(0)}{l}= \frac{1}{l}$.

(If the parentheses were around the entire $\frac{n\pi x}{l}$ then the value would be $cos\left(\frac{n\pi 0}{l}\right)= cos(0)= 1$.)

Answer 2

Here we assume $l\neq 0$. If $x=0$, we have that $$ \cos(n\pi\times 0)/l=\cos(0)/l=1/l. $$ If you mean $\cos(n\pi x/l)$, then, at $x=0$, we have $$ \cos(n\pi\times 0/l)=\cos(0)=1. $$