How do you prove \[\tan \left( 180{}^\circ +a \right)=\tan \left( a \right)\]

Hint: Here in the given question we have to prove the given following for that we must use here the fundamental trigonometric identities for proving the given term. As the given turn is in $\tan \left( a+b \right)$ form. We have the identity of $\tan \left( a+b \right)=\tan a+\tan b$
$=1-\tan a.\tan b$
To prove the given term. There are so many trigonometric identities which are used in various problems. We have to remember all their identities.

Complete step by step solution:
Here we have,
$\tan \left( 180{}^\circ +a \right)$
But we have to prove here $\tan \left( 180{}^\circ +a \right)=\tan \left( a \right)$
So, as we seen the problem it shows that it is a same as the $\tan \left( a+b \right)$
This is the trigonometric identity. There are also various identities.
The trigonometric Identity for
$\tan \left( a+b \right)=\dfrac{\tan a+\tan b}{1-\tan a-\tan b}$
So, Here $b=180{}^\circ $ and $\tan 180{}^\circ =0$
You have to remembers one point that the period of $\sin \left( t \right)$ and $\cos \left( t \right)$ is $2\pi =360{}^\circ $ and
In case of $\tan \left( t \right)$ It is $\pi =180{}^\circ $
Hence,
It is proved that $\tan \left( 180{}^\circ +a \right)=\tan \left( a \right)$

Additional Information:
The trigonometric identities are that which involves trigonometric functions in which both sides of equality are defined. These identities are very careful when any expression which involves trigonometric functions needs to be simplified. There are many techniques, While solving any problem using the substitution rule with a trigonometric function, and simplifying the resulting integral with a trigonometric identity. By using the identity we will prove the problems in there we have two sides of the equation, one is left hand and the other side is right one and to prove the identity we need to use logical steps showing that one side of the equation ends up with the other side of the equation.

Note: While using identities check the sign what is in the given problem because there are various identities for various problems? The trigonometric involved in an equation. There are six categories of trigonometric identities. Each of this is a key trigonometric identity and we should memorize it. While verifying and proving any question we should use the identity which makes two sides of a given equation identical in order to prove that it is true. These are some key points you should remember while solving any problem of trigonometry.