Question

What is the value of \[{\sin ^2}{60^ \circ } \cdot {\cos ^2}{30^ \circ } + \tan {45^ \circ } \cdot \cos {60^ \circ }\sin {30^ \circ }\]?
(A) \[\dfrac{{13}}{{16}}\]
(B) \[5\]
(C) \[1\]
(D) \[ - \dfrac{1}{2}\]

Answer 1

Hint: According to this question, we need to remember the values of the angles from the trigonometric table. This table is used when we want to know the values of the trigonometric standard angles. After that we can put the values in place of the angles and solve accordingly.

Complete step-by-step solution:
The given trigonometric identity is:
\[{\sin ^2}{60^ \circ } \cdot {\cos ^2}{30^ \circ } + \tan {45^ \circ } \cdot \cos {60^ \circ }\sin {30^ \circ }\]
We have to first put the values of the angles. For that we should know the trigonometry table. The trigonometry table contains the trigonometric formulas for the angles. The values of the common angles are given in the table.
We know that the value of \[\sin {60^ \circ }\]is \[\dfrac{{\sqrt 3 }}{2}\], the value of \[\cos {30^ \circ }\]is also \[\dfrac{{\sqrt 3 }}{2}\], the value of \[\tan {45^ \circ }\]is \[1\], \[\cos {60^ \circ }\]is \[\dfrac{1}{2}\], and the value of \[\sin {30^ \circ }\]is also \[\dfrac{1}{2}\]. Now, we will put these values in the given question, and we get:
\[ = {\left( {\dfrac{{\sqrt 3 }}{2}} \right)^2} \cdot {\left( {\dfrac{{\sqrt 3 }}{2}} \right)^2} + 1 \cdot \left( {\dfrac{1}{2}} \right)\left( {\dfrac{1}{2}} \right)\]
Now, we will open the brackets, and we get:
\[ = \left( {\dfrac{3}{4}} \right) \cdot \left( {\dfrac{3}{4}} \right) + 1 \cdot \left( {\dfrac{1}{2}} \right)\left( {\dfrac{1}{2}} \right)\]
Now, we will multiply the given terms, and we get:
\[ = \dfrac{9}{{16}} + 1 \cdot \dfrac{1}{4}\]
\[ = \dfrac{9}{{16}} + \dfrac{1}{4}\]
Now, we will add the two terms or we can say that we will add the two fractions. We will take the LCM here as 16, and we get:
\[ = \dfrac{{9 + 4}}{{16}}\]
Now, here we will add the numerator, and we get:
\[ = \dfrac{{13}}{{16}}\]
This is our final answer.

Therefore, we can say that the value for the trigonometric identity \[{\sin ^2}{60^ \circ } \cdot {\cos ^2}{30^ \circ } + \tan {45^ \circ } \cdot \cos {60^ \circ }\sin {30^ \circ }\]is \[\dfrac{{13}}{{16}}\]. So, option (a) is correct.


Note: We need to always use the correct formula at the correct place. There are many places where we can use more than one formula, but we have to see which formula is more suitable. Choosing other formulas may also lead to an answer, but they may not be the desired answers.