Question

Find the value of the expression,
\[\left( \cos 0{}^\circ +\sin 45{}^\circ +\sin 30{}^\circ \right)\left( \sin 90{}^\circ -\cos 45{}^\circ +\cos 60{}^\circ \right)\] .
(A) \[\dfrac{5}{4}\]
(B) \[\dfrac{9}{4}\]
(C) \[\dfrac{7}{4}\]
(D) None of these

Answer 1

Hint: We have the expression, \[\left( \cos 0{}^\circ +\sin 45{}^\circ +\sin 30{}^\circ \right)\left( \sin 90{}^\circ -\cos 45{}^\circ +\cos 60{}^\circ \right)\] . Here, we have sine and cosine terms. We know that \[\sin 45{}^\circ =\dfrac{1}{\sqrt{2}}\] , \[\cos 45{}^\circ =\dfrac{1}{\sqrt{2}}\] , \[\cos 60{}^\circ =\dfrac{1}{2}\] , \[\cos 0{}^\circ =1\] , \[\sin 90{}^\circ =1\] , and \[\sin 30{}^\circ =\dfrac{1}{2}\] . Put these values in the expression and calculate the value of \[x\] .

Complete step by step answer:
According to the question, we are given an expression in which there are trigonometric terms and we have to find the value of the expression.
The given expression is \[\left( \cos 0{}^\circ +\sin 45{}^\circ +\sin 30{}^\circ \right)\left( \sin 90{}^\circ -\cos 45{}^\circ +\cos 60{}^\circ \right)\] …………………………………(1)
In the above equation, we can observe that the expression is having sine terms and cosine terms.
The expression is somewhat complex which needs to be simplified into a simpler form.
We need the values of sine and cosine terms to calculate the value of the given expression.
We know the value that \[\sin 45{}^\circ =\dfrac{1}{\sqrt{2}}\] ………………………………………(2)
We also know the value that \[\cos 45{}^\circ =\dfrac{1}{\sqrt{2}}\] ………………………………………….(3)
We also know the value that \[\sin 30{}^\circ =\dfrac{1}{2}\] …………………………………………..(4)
We also know the value that \[\cos 60{}^\circ =\dfrac{1}{2}\] …………………………………………..(5)
We also know the value that \[\cos 0{}^\circ =1\] …………………………………………..(6)
We also know the value that \[\sin 90{}^\circ =1\] …………………………………………..(7)
We have to use equation (2), equation (3), equation (4), equation (5), equation (6) and equation (7) to simplify equation (1).
Now, in equation (1), on substituting \[\sin 45{}^\circ \] by \[\dfrac{1}{\sqrt{2}}\] , \[\cos 45{}^\circ \] by \[\dfrac{1}{\sqrt{2}}\] , \[\sin 30{}^\circ \] by \[\dfrac{1}{2}\] , \[\cos 60{}^\circ \] by \[\dfrac{1}{2}\] , \[\sin 90{}^\circ \] by 1 and, \[\cos 0{}^\circ \] by 1, we get
 \[=\left( 1+\dfrac{1}{\sqrt{2}}+\dfrac{1}{2} \right)\left( 1-\dfrac{1}{\sqrt{2}}+\dfrac{1}{2} \right)\] …………………………………(8)
On simplifying the above equation, we get
\[=\left( \dfrac{3}{2}+\dfrac{1}{\sqrt{2}} \right)\left( \dfrac{3}{2}-\dfrac{1}{\sqrt{2}} \right)\]
Now, on multiplying and simplifying, we get
\[\begin{align}
  & =\dfrac{9}{4}-\dfrac{3}{2\sqrt{2}}+\dfrac{3}{2\sqrt{2}}-\dfrac{1}{2} \\
 & =\dfrac{9}{4}-\dfrac{1}{2} \\
 & =\dfrac{9-2}{4} \\
\end{align}\]
\[=\dfrac{7}{4}\] …………………………………….(9)
From equation (9), we have the value of the expression.
Therefore, the value of the expression \[\left( \cos 0{}^\circ +\sin 45{}^\circ +\sin 30{}^\circ \right)\left( \sin 90{}^\circ -\cos 45{}^\circ +\cos 60{}^\circ \right)\] is \[\dfrac{7}{4}\] and option (C) is the correct answer.

Note:
 In this type of question where we have an expression in which we have trigonometric terms and we are asked to find the value of the expression. Always approach this type of question by putting the values of sine and cosine terms.