Evaluate \[\sin 60^\circ \cos 30^\circ + \sin 30^\circ \cos 60^\circ \]. What is the value of \[\sin \left( {60^\circ + 30^\circ } \right)\]. What can you conclude?

Hint: Here we will put the values of the sine and cosine functions in the first expression and find the required value. We will then find the values of the second expression by adding the angles and substituting the value of sine function of the obtained angle. Then we will compare the values of both the expressions. Then by using this we will get the required relationship between the given expressions.

Complete step by step solution:
First, we will find the value of the given expression i.e. \[\sin 60^\circ \cos 30^\circ + \sin 30^\circ \cos 60^\circ \].
We know that the value of \[\sin 30^\circ = \dfrac{1}{2},\sin 60^\circ = \dfrac{{\sqrt 3 }}{2},\cos 30^\circ = \dfrac{{\sqrt 3 }}{2},\cos 60^\circ = \dfrac{1}{2}\].
Now substituting all these values in the expression, we get
\[\sin 60^\circ \cos 30^\circ + \sin 30^\circ \cos 60^\circ = \dfrac{{\sqrt 3 }}{2} \times \dfrac{{\sqrt 3 }}{2} + \dfrac{1}{2} \times \dfrac{1}{2}\]
Multiplying the terms, we get
\[ \Rightarrow \sin 60^\circ \cos 30^\circ + \sin 30^\circ \cos 60^\circ = \dfrac{3}{4} + \dfrac{1}{4} = 1\]…………………….\[\left( 1 \right)\]
Now we will find the value of \[\sin \left( {60^\circ + 30^\circ } \right)\]. Therefore, by simply adding the angle inside the bracket we can write the equation as
\[\sin \left( {60^\circ + 30^\circ } \right) = \sin \left( {90^\circ } \right)\]
We know that the value of \[\sin \left( {90^\circ } \right) = 1\]. Therefore, we get
\[ \Rightarrow \sin \left( {60^\circ + 30^\circ } \right) = 1\]……………………..\[\left( 2 \right)\]
Now we can see from the equation \[\left( 1 \right)\] and equation \[\left( 2 \right)\] that the values as these equations are equations. Therefore, we get
\[ \Rightarrow \sin \left( {60^\circ + 30^\circ } \right) = \sin 60^\circ \cos 30^\circ + \sin 30^\circ \cos 60^\circ \]

Hence from this we get this identity as
\[ \Rightarrow \sin \left( {A + B} \right) = \sin A\cos B + \cos A\sin B\]

Note:
In order to solve this question, we need to know the different values of the trigonometric function for different angles. Also in which quadrant which function is positive or negative. In the first quadrant, all the functions i.e. sin, cos, tan, cot, sec, cosec are positive. In the second quadrant, only the sin and cosec function are positive and all the other functions are negative. In the third quadrant, only tan and cot function is positive and in the fourth quadrant, only cos and sec function is positive.