Question

How do you find the value of \[\cot \left( {\dfrac{\pi }{4}} \right)\] ?

Answer 1

Hint: In order to solve this question, we can proceed by finding the \[\sin \] and \[\cos \] of the same angle as given and we know that \[\cot x = \dfrac{1}{{\tan x}}\] and \[\tan x = \dfrac{{\sin x}}{{\cos x}}\] .Therefore we divide both the values of \[\sin \] and \[\cos \] of the same angle to find the value of \[\tan \] and then reciprocate the value to get the value of \[\cot \] for the same angle i.e., \[\dfrac{\pi }{4}\] and get our desired result.

Complete step-by-step answer:
Now we are given a trigonometric function, \[\cot \left( {\dfrac{\pi }{4}} \right)\]
and we have to find the value of \[\cot \left( {\dfrac{\pi }{4}} \right)\]
So, we know that,
The value of \[\sin \left( {\dfrac{\pi }{4}} \right)\] is equals to \[\dfrac{1}{{\sqrt 2 }}\]
And the value of \[\cos \left( {\dfrac{\pi }{4}} \right)\] is also equals to \[\dfrac{1}{{\sqrt 2 }}\]
Now using trigonometric identity
i.e., \[\tan x = \dfrac{{\sin x}}{{\cos x}}\]
First, we will find out the value of \[\tan \left( {\dfrac{\pi }{4}} \right)\]
Therefore, \[\tan \left( {\dfrac{\pi }{4}} \right) = \dfrac{{\sin \left( {\dfrac{\pi }{4}} \right)}}{{\cos \left( {\dfrac{\pi }{4}} \right)}}\]
On substituting the values of \[\sin \left( {\dfrac{\pi }{4}} \right)\] and \[\cos \left( {\dfrac{\pi }{4}} \right)\]
\[\tan \left( {\dfrac{\pi }{4}} \right) = \dfrac{{\dfrac{1}{{\sqrt 2 }}}}{{\dfrac{1}{{\sqrt 2 }}}}\]
On dividing, we get
\[ \Rightarrow \tan \left( {\dfrac{\pi }{4}} \right) = 1\]
Now again using trigonometric identity,
i.e., \[\cot x = \dfrac{1}{{\tan x}}\]
we will now find out the value of \[\cot \left( {\dfrac{\pi }{4}} \right)\]
So, \[\cot \left( {\dfrac{\pi }{4}} \right) = \dfrac{1}{{\tan \left( {\dfrac{\pi }{4}} \right)}}\]
On substituting the value of \[\tan \left( {\dfrac{\pi }{4}} \right)\] ,we get
\[\cot \left( {\dfrac{\pi }{4}} \right) = \dfrac{1}{1}\]
\[ \Rightarrow \cot \left( {\dfrac{\pi }{4}} \right) = 1\]
which is the required answer.
Hence, the value of \[\cot \left( {\dfrac{\pi }{4}} \right)\] is equal to \[1\]
So, the correct answer is “1”.

Note: Here in these types of problems where we are asked to find the value of the cotangent or tangent of any angle, we must know the basic values of the sine and cosine of the angle like \[0^\circ ,{\text{ }}30^\circ ,{\text{ }}45^\circ ,{\text{ }}60^\circ ,{\text{ }}90^\circ \] which can also be written as \[0^\circ ,{\text{ }}\dfrac{\pi }{6},{\text{ }}\dfrac{\pi }{4},{\text{ }}\dfrac{\pi }{3},{\text{ }}\dfrac{\pi }{2}\] and then we can easily calculate the same angles of any given function.
Also, there is an alternative way to solve this question i.e.,
As we know that,
\[\cot x = \dfrac{1}{{\tan x}}{\text{ }} - - - \left( 1 \right)\]
And \[\tan x = \dfrac{{\sin x}}{{\cos x}}{\text{ }} - - - \left( 2 \right)\]
So, from \[\left( 1 \right)\] and \[\left( 2 \right)\] we can write
\[\cot x = \dfrac{1}{{\dfrac{{\sin x}}{{\cos x}}}}{\text{ }}\]
\[ \Rightarrow \cot x = \dfrac{{\cos x}}{{\sin x}}{\text{ }}\]
So, here we can directly put the values of \[\sin \] and \[\cos \] at the same angle to get the result.
As we know that,
The value of \[\sin \left( {\dfrac{\pi }{4}} \right)\] is equals to \[\dfrac{1}{{\sqrt 2 }}\]
And the value of \[\cos \left( {\dfrac{\pi }{4}} \right)\] is also equals to \[\dfrac{1}{{\sqrt 2 }}\]
Therefore, the value of \[\cot \left( {\dfrac{\pi }{4}} \right)\] will be equals to
\[\cot \left( {\dfrac{\pi }{4}} \right) = \dfrac{{\cos \left( {\dfrac{\pi }{4}} \right)}}{{\sin \left( {\dfrac{\pi }{4}} \right)}}{\text{ }}\]
On substituting values, we get
\[\cot \left( {\dfrac{\pi }{4}} \right) = \dfrac{{\dfrac{1}{{\sqrt 2 }}}}{{\dfrac{1}{{\sqrt 2 }}}}{\text{ }}\]
\[ \Rightarrow \cot \left( {\dfrac{\pi }{4}} \right) = 1{\text{ }}\]
Hence, we get our required result.
i.e., the value of \[\cot \left( {\dfrac{\pi }{4}} \right)\] is equal to \[1\]