Question

What is the value of the trigonometric expression?
$\sin {45^o}\cos {45^0}{\left( {\tan {{45}^o} + \cot {{45}^0}} \right)^2}$
$
  (a){\text{ 1}} \\
  (b){\text{ 2}} \\
  (c){\text{ 3}} \\
  (d){\text{ 4}} \\
 $

Answer 1

Hint – There can be two ways to approach this problem, one is the direct way, in this we will be simply using the value of standard trigonometric ratios that is $\tan {45^o} = \cot {45^o} = 1$ and $\sin {45^o} = \cos {45^o} = \dfrac{1}{{\sqrt 2 }}$, substitution of these values directly into the trigonometric expression and the simplification will help getting the answer. The second method we will discuss at the end of this solution.

Complete step-by-step answer:
Given trigonometric expression is
$\sin {45^o}\cos {45^0}{\left( {\tan {{45}^o} + \cot {{45}^0}} \right)^2}$
As we know that $\tan {45^o} = \cot {45^o} = 1$
So substitute this value in above equation we have,
$ \Rightarrow \sin {45^o}\cos {45^0}{\left( {1 + 1} \right)^2}$
$ \Rightarrow \sin {45^o}\cos {45^0}{\left( 2 \right)^2}$
$ \Rightarrow 4\sin {45^o}\cos {45^0}$
Now as we know that $\sin {45^o} = \cos {45^o} = \dfrac{1}{{\sqrt 2 }}$
So substitute this value in above equation we have,
$ \Rightarrow 4\left( {\dfrac{1}{{\sqrt 2 }}} \right)\left( {\dfrac{1}{{\sqrt 2 }}} \right) = \dfrac{4}{2} = 2$
So this is the required answer of the given expression.
Hence option (B) is the correct answer.

Note – In the second method first we will be simplifying the given trigonometric expression, so to the expression of $\sin {45^o}\cos {45^0}{\left( {\tan {{45}^o} + \cot {{45}^0}} \right)^2}$ apply the algebraic identity of ${(a + b)^2} = {a^2} + {b^2} + 2ab$. Then after simplification we will be substituting the value of standard ratios, this method will be lengthy and will yield the same answer thus it is advised to go for the first method only.