Question

If \[x = {\cos ^2}\theta \] and \[y = \cot \theta \], then find \[\dfrac{{dy}}{{dx}}\]at \[\theta = \dfrac{\pi }{4}\]

Answer 1

Hint: Here, we will use the concept of differentiation to find the value of \[\dfrac{{dy}}{{dx}}\]. We will first find the value of the \[dx\] and then we will find the value of the \[dy\] separately. Then by dividing their values we will get the value of \[\dfrac{{dy}}{{dx}}\]. Further, we will substitute the value of \[\theta = \dfrac{\pi }{4}\] to get the final answer.

Complete step-by-step answer:
We will first find the value of \[dx\] by differentiating the equation ,\[x = {\cos ^2}\theta \].
We know that the differentiation of \[\cos \theta \] is \[ - \sin \theta \].
Differentiating the equation \[x = {\cos ^2}\theta \], we get
\[dx = - 2\cos \theta \sin \theta \,d\theta \]……………….\[\left( 1 \right)\]
Now we will find the value of \[dy\] by differentiating the equation \[y = \cot \theta \].
We know that the differentiation of \[\cot \theta \] is \[ - \cos e{c^2}\theta \]. So, we get
\[dy = - \cos e{c^2}\theta \,d\theta \]……………….\[\left( 2 \right)\]
Now by dividing the equation \[\left( 2 \right)\] by \[\left( 1 \right)\], we get
\[\dfrac{{dy}}{{dx}} = \dfrac{{ - \cos e{c^2}\theta \,d\theta }}{{ - 2\cos \theta \sin \theta \,d\theta }}\]
We know that \[{\rm{cosec}}\theta \] is the reciprocal of the \[\sin \theta \]. Therefore, we get
\[ \Rightarrow \dfrac{{dy}}{{dx}} = \dfrac{1}{{2{{\sin }^2}\theta \cos \theta \sin \theta }}\]
Now we will find the value of the \[\dfrac{{dy}}{{dx}}\] at \[\theta = \dfrac{\pi }{4}\].
We know that the value of \[\sin \dfrac{\pi }{4} = \dfrac{1}{{\sqrt 2 }}\] and \[\cos \dfrac{\pi }{4} = \dfrac{1}{{\sqrt 2 }}\].
Substituting \[\sin \dfrac{\pi }{4} = \dfrac{1}{{\sqrt 2 }}\] and \[\cos \dfrac{\pi }{4} = \dfrac{1}{{\sqrt 2 }}\] in the equation, we get
\[ \Rightarrow \dfrac{{dy}}{{dx}} = \dfrac{1}{{2 \times {{\left( {\dfrac{1}{{\sqrt 2 }}} \right)}^2} \times \dfrac{1}{{\sqrt 2 }} \times \dfrac{1}{{\sqrt 2 }}}}\]
Simplifying the equation, we get
\[ \Rightarrow \dfrac{{dy}}{{dx}} = \dfrac{1}{{2 \times \dfrac{1}{2} \times \dfrac{1}{2}}} = 2\]
Hence, \[\dfrac{{dy}}{{dx}}\] at \[\theta = \dfrac{\pi }{4}\] is 2.

Note: Here, we need to know the basic differentiation of the trigonometric function in order to solve questions. We have used differentiation by parts to find the value of \[dx\]. Differentiation is a method by which we can measure per unit of a function in the given independent variable.