Prove that: \[\sin {{780}^{\circ }}\sin {{120}^{\circ }}+\cos {{240}^{\circ }}\sin {{390}^{\circ }}=\dfrac{1}{2}\]

Hint: We will first simplify the terms in the left hand side of the expression in multiples of 90 plus some angle and then with the help of cofunction identities we will convert these terms into simple functions with standard angles and after substituting we will solve this question.

Complete step-by-step answer:

The given expression in the question is \[\sin {{780}^{\circ }}\sin {{120}^{\circ }}+\cos {{240}^{\circ }}\sin {{390}^{\circ }}=\dfrac{1}{2}......(1)\]

Now we will start with the left hand side of the equation (1). So.

\[\Rightarrow \sin {{780}^{\circ }}\sin {{120}^{\circ }}+\cos {{240}^{\circ }}\sin {{390}^{\circ }}......(2)\]

Now transform all the angles in equation (2) into a sum of 90 or 90 multiple and some standard angle.

So we can write 780 as 90 multiplied by 8 plus 60.

And we can write 120 as 90 multiplied plus 30.

Also we can write 240 as 90 multiplied by 3 minus 30.

We can also write 390 as 90 multiplied by 4 plus 30.

So now using all this information in equation (2) we get,

\[\Rightarrow \sin (8\times {{90}^{\circ }}+{{60}^{\circ }})\sin ({{90}^{\circ }}+{{30}^{\circ }})+\cos (3\times {{90}^{\circ }}-{{30}^{\circ }})\sin (4\times {{90}^{\circ }}+{{30}^{\circ }})......(3)\]

Now applying cofunction identities that is \[\sin (90+\theta )=\cos \theta \] and \[\sin \left( n\dfrac{\pi }{2}+\theta \right)=\sin \theta \] and \[\cos \left( n\dfrac{\pi }{2}-\theta \right)=-\sin \theta \] in equation (3) we get,

\[\Rightarrow \sin {{60}^{\circ }}\cos {{30}^{\circ }}-\sin {{30}^{\circ }}\sin {{30}^{\circ }}......(4)\]

Now we know the value of all the standard angles and hence substituting them in equation (4) we get,

\[\Rightarrow \dfrac{\sqrt{3}}{2}\times \dfrac{\sqrt{3}}{2}-\dfrac{1}{2}\times \dfrac{1}{2}......(5)\]

Now solving equation (5) we get,

\[\Rightarrow \dfrac{3}{4}-\dfrac{1}{4}=\dfrac{1}{2}\]

Hence this is equal to the right hand side of the equation. Hence proved.

Note: In trigonometry remembering the formulas and the identities is very important because then it becomes easy. We in a hurry can make a mistake in applying the cofunction identities as we can write cos in place of sin and sin in place of cos in equation (4). Also if we don’t remember the formula then we can get confused about how to proceed further after equation (3).