Question

The value of \[\cot 22{{\dfrac{1}{2}}^{\circ }}\] is equal to?
(a) \[1+\dfrac{1}{\sqrt{2}}\]
(b) \[1+\sqrt{2}\]
(c) \[\sqrt{2}-1\]
(d) None of these

Answer 1

Hint: For solving this question you should know about the general formulas of trigonometrics. These problems are solved by converting a trigonometric function to another function and then solve this to get the answer. In this problem we will solve this by changing the \[\cot \theta \] in the form of \[\dfrac{\cos \theta }{\sin \theta }\]. And then we use the formula of \[\sin 2A\] and \[\cos 2A\] for solving this. Thus, we will find the answer for this.

Complete step by step answer:
According to the question it is asked to find the value of \[\cot 22{{\dfrac{1}{2}}^{\circ }}\].
So, as we know that the value of \[\pi ={{180}^{\circ }}\] and \[\dfrac{\pi }{2}\] is equal to \[{{90}^{\circ }}\] and \[\dfrac{\pi }{4}\] will be \[{{45}^{\circ }}\] and the value of \[\dfrac{\pi }{8}\] is equals to \[22.{{\dfrac{1}{2}}^{\circ }}\].
So, here we consider that \[\theta \] is equal to \[{{45}^{\circ }}\].
Let \[\theta ={{45}^{\circ }}\]
And \[\dfrac{\theta }{2}=\dfrac{45}{2}=22{{\dfrac{1}{2}}^{\circ }}\]
As we know that the \[\cot X=\dfrac{\cos X}{\sin X}\]
So, here we can write \[\cot \dfrac{\theta }{2}=\dfrac{\cos \dfrac{\theta }{2}}{\sin \dfrac{\theta }{2}}\]
And we can write it as,
\[\cot \dfrac{\theta }{2}=\dfrac{\left[ 2\cos \dfrac{\theta }{2}\cos \dfrac{\theta }{2} \right]}{\left[ 2\sin \dfrac{\theta }{2}\cos \dfrac{\theta }{2} \right]}\]
Using the formulas \[\sin 2A=2\sin A\cos A\] and \[\cos A=2{{\cos }^{2}}A-1\]
\[\cos A=\dfrac{\left[ 2{{\cos }^{2}}\left( \dfrac{\theta }{2} \right) \right]}{\sin \theta }\]
            \[=\dfrac{\left( 1+\cos \theta \right)}{\sin \theta }\]
Now, substituting the values of \[\theta \] and \[\dfrac{\theta }{2}\], we get,
\[\cot 22{{\dfrac{1}{2}}^{\circ }}=\dfrac{\left( 1+\cos {{45}^{\circ }} \right)}{\sin {{45}^{\circ }}}\]
                   \[\begin{align}
  & =\dfrac{\left[ 1+\dfrac{1}{\sqrt{2}} \right]}{\dfrac{1}{\sqrt{2}}} \\
 & =\sqrt{2}+1 \\
\end{align}\]

So, the correct answer is “Option b”.

Note: While solving these type of questions you have to be careful about the formulas because if we use another formula at other place then it may be possible that our question will be solved but it will take too much time and in the most of the cases it can’t be solved.