Question

The value of $\sin 105^\circ .\sin 75^\circ $ is
(A). $\dfrac{{\sqrt 3 + 2}}{4}$
(B). $\dfrac{{\sqrt 3 + 2}}{2}$
(C). $\dfrac{{\sqrt 3 - 2}}{4}$
(D). $\dfrac{{\sqrt 3 - 2}}{2}$

Answer 1

Hint: In this particular type of question proceed by converting the given into the form of $\sin \left( {a + b} \right).\sin \left( {a - b} \right)$ and then expand using the rules of trigonometry . Cancel out the common terms and simplify to get the desired answer.

Complete step-by-step answer:

$\sin 105^\circ .\sin 75^\circ $
$
   = \sin \left( {90 + 15} \right).\sin \left( {90 - 15} \right) \\
   = \left( {\sin 90.\cos 15 + \cos 90.\sin 15} \right) \times \left( {\sin 90.\cos 15 - \cos 90.\sin 15} \right) \\
   = \cos 15 \times \cos 15 = {\cos ^2}15 \\
$
 (since $\sin \left( {a \pm b} \right) = \sin a.\cos b \pm \cos a.\sin b$ )
We know that cos15 =$\dfrac{{\sqrt {2 + \sqrt 3 } }}{2}$
Therefore , ${\cos ^2}15 = {\left( {\dfrac{{\sqrt {2 + \sqrt 3 } }}{2}} \right)^2} = \dfrac{{2 + \sqrt 3 }}{4}$

Note: Remember that the value of cos15 can be in a different form if other methods to find its value are used . Recall all the formulas of trigonometric functions to solve such types of questions accurately . Also note that there are other methods like multiplying and expanding using the FOIL method to solve this particular type of question.