How do we simplify this expression \[\sin \left( {4\pi + x} \right)\] ?

Hint: We use the angle sum formula of sin to simplify the given expression. The Trigonometric identities are Basic and Pythagorean identities, Angle sum and difference identities, Double angle identities, Half angle identities, Product identities, sum identities.
Some of the Angle sum and difference identities are :-
 \[\sin \left( {A + B} \right) = \sin A\cos B + \cos A\sin B\]
 \[\sin \left( {A - B} \right) = \sin A\cos B - \cos A\sin B\]
\[\cos \left( {A + B} \right) = \cos A\cos B - \sin A\sin B\]
 \[\tan \left( {A + B} \right) = \dfrac{{\tan A + \tan B}}{{1 - \tan A\tan B}}\]

Complete step-by-step answer:
In this question we have been asked to simplify \[\sin (4\pi + x)\]
According to angle sum and difference identities
 \[\sin \left( {A + B} \right) = \sin A\cos B + \cos A\sin B\]
So, now applying this rule on \[\sin (4\pi + x)\] , we get
 \[\sin (4\pi + x) = \sin 4\pi \cos x + \cos 4\pi \sin x\] \[\_\_\_\_\_\left( 1 \right)\]
Now, we know the value of \[\sin 4\pi = 0\] and the value of \[\cos 4\pi = 1\] .
So, now putting the value of \[\sin 4\pi \] and \[\cos 4\pi \] in equation \[\left( 1 \right)\] , we get,
\[\sin (4\pi + x) = 0 \times \cos x + 1 \times \sin x\]
\[\sin (4\pi + x) = 0 + \sin x\]
\[\sin (4\pi + x) = \sin x\]
So finally on simplifying the expression of \[\sin (4\pi + x)\] , we get \[\sin x\] .
\[\therefore \sin (4\pi + x) = \sin x\] is the required answer.
So, the correct answer is “sin x”.

Note: Trigonometric identities are equations that relate different trigonometric functions. Trigonometric identities are used to solve different trigonometric and geometric problems and understand various mathematical properties. The three main functions in trigonometry are Sine, Cosine and tangent.
One should know the correct values of trigonometric identities and should avoid casual mistakes which generally take place on missing out the negative signs.
The values of \[\sin \pi = 0\] and \[\cos \pi = - 1\] . You should learn all the formulas regarding trigonometric identities and their sum and difference identities.