<s>[INST] How can I find the limit of given trigonometric function? I am messed up on solving this question. What should I do first in order to get the answer ?
This is the trigonometric function
$$ \lim \limits_{x \rightarrow 0} \frac{(a+x)\sec(a+x) - a \sec(a)}{x} $$
 [/INST] Changing into cosines greatly eases the manipulation of terms.
$$
\begin{align} 
\lim \limits_{x \rightarrow 0} \frac{(a+x)\sec(a+x) - a \sec(a)}{x} 
& = \lim \limits_{x \rightarrow 0} \frac{a\sec(a+x) - a \sec(a)}{x} + \lim \limits_{x \rightarrow 0} \frac{x\sec(a+x)}{x} \\
& = A + B
\end{align}
$$
$$
\begin{align}
A & = \lim \limits_{x \rightarrow 0} \frac{a\sec(a+x) - a \sec(a)}{x} \\
& = a\lim \limits_{x \rightarrow 0} \frac{\cos(a) - \cos(a+x)}{\cos(a)\cos(a+x)x} \\
& = a\lim \limits_{x \rightarrow 0} 2\frac{\sin(a + x/2) sin(x/2)}{\cos(a)\cos(a+x)x} \\
& = a\lim \limits_{x \rightarrow 0} \; \frac{\sin(a + x/2)}{\cos(a)\cos(a+x)} \frac {\sin(x/2)}{x/2} \\
& = a\tan(a)sec(a)
\end{align}
$$
$$
B = \sec(a)
$$
</s>