Answer

Verified

395.7k+ views

**Hint**: First simplify the given equation by substituting $ x = \tan \theta $ . After simplification, use the range of $ \sin \theta $ to find the range of the given expression. You can use the fact that $ {\tan ^{ - 1}}x $ is an increasing function. So the inequality will not change.

**:**

__Complete step-by-step answer__The given equation is

$ \Rightarrow y = {\tan ^{ - 1}}\left( {\dfrac{{2x}}{{1 + {x^2}}}} \right) $

To simplify this equation, put $ x = \tan \theta $ . Then

$ \dfrac{{2x}}{{1 + {x^2}}} = \dfrac{{2\tan \theta }}{{1 + {{\tan }^2}\theta }} $

We have a formula,

$ \dfrac{{2\tan \theta }}{{1 + {{\tan }^2}\theta }} = \sin 2\theta $

By using this formula, we can write

$ \dfrac{{2x}}{{1 + {x^2}}} = \sin 2\theta $

Now, we know that the range of $ \sin 2\theta $ is $ \left[ { - 1,1} \right] $ . Because the maximum value of $ \sin 2\theta $ is 1 and its minimum value is -1. Also it is a continuous function. So it takes all the values between -1 and 1.

Thus, $ \dfrac{{2x}}{{1 + {x^2}}} \in [ - 1,1] $

Now, by applying $ {\tan ^{ - 1}} $ to both the sides. And knowing that $ {\tan ^{ - 1}} $ is an increasing function. So the inequality in intervals will not change.

$ \therefore {\tan ^{ - 1}}\left( {\dfrac{{2x}}{{1 + {x^2}}}} \right) \in \left[ {{{\tan }^{ - 1}}( - 1),{{\tan }^{ - 1}}(1)} \right] $

By using the property, $ {\tan ^{ - 1}}( - x) = {-\tan ^{ - 1}}x $ , we can write

$ = \left[ { - {{\tan }^{ - 1}}1,{{\tan }^{ - 1}}1} \right] $

We know that, $ {\tan ^{ - 1}}1 = \dfrac{\pi }{4} $

Thus, we get the range as

$ \Rightarrow {\tan ^{ - 1}}\left( {\dfrac{{2x}}{{1 + {x^2}}}} \right) \in \left[ { - \dfrac{\pi }{4},\dfrac{\pi }{4}} \right] $

Therefore, from the above explanation, the correct answer is, option (A) $ \left[ { - \dfrac{\pi }{4},\dfrac{\pi }{4}} \right] $

**So, the correct answer is “Option A”.**

**Note**: To solve this question, you need to know the trigonometric formulae. Then only it would click you that you can simplify the equation in terms of $ \sin \theta $ . The key point here is to know the range of the sine function and know that the inequality does not change when you apply an increasing function to it. If you check the graph of $ {\tan ^{ - 1}}x $ . You will observe that, it is an increasing function.

Recently Updated Pages

Cryolite and fluorspar are mixed with Al2O3 during class 11 chemistry CBSE

Select the smallest atom A F B Cl C Br D I class 11 chemistry CBSE

The best reagent to convert pent 3 en 2 ol and pent class 11 chemistry CBSE

Reverse process of sublimation is aFusion bCondensation class 11 chemistry CBSE

The best and latest technique for isolation purification class 11 chemistry CBSE

Hydrochloric acid is a Strong acid b Weak acid c Strong class 11 chemistry CBSE

Trending doubts

Which are the Top 10 Largest Countries of the World?

Define limiting molar conductivity Why does the conductivity class 12 chemistry CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Difference Between Plant Cell and Animal Cell

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Name 10 Living and Non living things class 9 biology CBSE

The Buddhist universities of Nalanda and Vikramshila class 7 social science CBSE

Write a letter to the principal requesting him to grant class 10 english CBSE