How do you simplify \[\sin \left( {{{90}^ \circ } + x} \right)\] ?

Hint: We have to simplify \[\sin \left( {{{90}^ \circ } + x} \right)\], so, we will use the sine sum identity as it will be the most suitable identity for this question to be solved. The identity used will be $\sin \left( {A + B} \right) = \sin A\cos B + \cos A\sin B$. By putting the values for $A$ and $B$ we can simplify \[\sin \left( {{{90}^ \circ } + x} \right)\].

Complete Step by Step Solution:
From the question, we know that we have to simplify \[\sin \left( {{{90}^ \circ } + x} \right)\], so, that we can understand this more easily and can use that identity in some other questions to solve.
As in the \[\sin \left( {{{90}^ \circ } + x} \right)\], ${90^ \circ }$ angle is added with $x$ so, we will use the sine sum identity which is $\sin \left( {A + B} \right) = \sin A\cos B + \cos A\sin B$
The above identity is most suitable for this question to be simplified. There is another identity which sine difference identity $\sin \left( {A - B} \right) = \sin A\cos B - \cos A\sin B$ , but in this identity $A$ is subtracted from $B$ which will not be compatible with \[\sin \left( {{{90}^ \circ } + x} \right)\].
Therefore, let $A = {90^ \circ }$ and $B = x$.
Putting these values in the identity $\sin \left( {A + B} \right) = \sin A\cos B + \cos A\sin B$ , we get –
$ \Rightarrow \sin \left( {{{90}^ \circ } + x} \right) = \sin {90^ \circ }\cos x + \cos {90^ \circ }\sin x$
Now, we know that, $\sin {90^ \circ } = 1$ and $\cos {90^ \circ } = 0$ . Therefore, putting these values in the above equation, we get –
$ \Rightarrow \sin \left( {{{90}^ \circ } + x} \right) = 1 \times \cos x + 0 \times \sin x$
Now, further solving, we get –
$ \Rightarrow \sin \left( {{{90}^ \circ } + x} \right) = \cos x$

Hence, \[\sin \left( {{{90}^ \circ } + x} \right)\] can be simplified as $\cos x$.

Note:
Here, first of all we should understand the question very well and what is asked in the question and then move into the right direction, so that we get our required result. Then, apply the suitable formula according to the question. Moreover, while simplifying one should be aware of the result and should avoid the calculation mistakes, so that they get their required result. Trigonometric identities should be kept in mind as they are required to solve any trigonometry question.
The identity $\sin \left( {{{90}^ \circ } + x} \right) = \cos x$ can be used to solve various other questions.