Simplify the following expression $\sin 420^\circ \cos 330^\circ + \cos \left( { - 300^\circ } \right)\sin 150^\circ = $

Hint – We will start solving this question by noting down the given trigonometric functions and their given angles. Then by adding and subtracting some associated angles of trigonometry and after the calculation process we will get the required result.

Complete step-by-step answer:

Here, it is given that,
$\sin 420^\circ \cos 330^\circ + \cos \left( { - 300^\circ } \right)\sin 150^\circ $ ………….. (1)
We have no values of $\sin 420^\circ ,\cos 330^\circ ,\cos \left( { - 300^\circ } \right)$ and $\sin 150^\circ $.
So, we will use some associated angles of trigonometry to find and value and get the required result.
First we will find the value of $\sin 420^\circ $, we get,
$\sin 420^\circ $
$ = \sin \left( {360^\circ + 60^\circ } \right)$
We know that $\sin \left( {360^\circ + \theta } \right) = \sin \theta $, therefore,
$
  \sin 60^\circ \\
   = \dfrac{{\sqrt 3 }}{2} \\
 $

Now, we will find the value of $\cos 330^\circ $, we obtain,
$\cos 330^\circ $
$ = \cos \left( {360^\circ - 30^\circ } \right)$
We know that $\cos \left( {360^\circ - \theta } \right) = \cos \theta $, therefore,
$
  \cos 30^\circ \\
   = \dfrac{{\sqrt 3 }}{2} \\
 $

Now, we will find the value of $\cos \left( { - 300^\circ } \right)$, we obtain,
$\cos \left( { - 300^\circ } \right)$
$ = \cos 300^\circ $
$ = \cos \left( {360^\circ - 60^\circ } \right)$
We know that $\cos \left( {360^\circ - \theta } \right) = \cos \theta $, therefore,
$
  \cos 60^\circ \\
   = \dfrac{1}{2} \\
 $

Now, we will find the value of $\sin 150^\circ $, we get,
$
  \sin 150^\circ \\
   = \sin \left( {180^\circ - 30^\circ } \right) \\
 $
We know that $\sin \left( {180^\circ - \theta } \right) = \sin \theta $, therefore.
$
  \sin 30^\circ \\
   = \dfrac{1}{2} \\
 $

Now,
$\sin 420^\circ \cos 330^\circ + \cos \left( { - 300^\circ } \right)\sin 150^\circ $$ = \dfrac{{\sqrt 3 }}{2} \times \dfrac{{\sqrt 3 }}{2} + \dfrac{1}{2} \times \dfrac{1}{2}$
  $
   = \dfrac{{{{\left( {\sqrt 3 } \right)}^2}}}{4} + \dfrac{1}{4} \\
   = \dfrac{3}{4} + \dfrac{1}{4} \\
   = \dfrac{{3 + 1}}{4} \\
   = \dfrac{4}{4} \\
   = 1 \\
 $

Note – Trigonometric angles are the angles given by the ratios of the trigonometric functions. The angle value ranges from 0-360 degrees. These questions are very simple and easy to solve, however, all the basic formulas and the values of the associated angles of trigonometry must be remembered.