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The value of\sin \cot ^{-1} \tan \cos ^{-1} x  is equal to

Option: 1

x


Option: 2

\frac{\pi}{2}


Option: 3

1


Option: 4

None of these


Answers (1)

best_answer

 

 

Graph of Principal Value of function f-1 (f (x)) (Part 2) -

Graph of Principal Value of function f-1 (f (x)) (Part 2)

  1. Graph of cot-1( cot (x))

Domain of the function is R and range is (0, π)

            y = cot-1( cot (x))

∴          cot y = cot x

⇒         y = nπ + x, n ∈ I (integer)  

 

To draw the graph of y = cot-1( cot (x)), we draw all the line of y = nπ + x, n ∈ I for 

Y ∈ (0, π).


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\text { Let } \cos ^{-1} x=\theta \\ \Rightarrow x=\cos \theta \\ \Rightarrow \tan \theta=\frac{\sqrt{1-x^{2}}}{x} \\ \sin \cot ^{-1} \tan \cos^{-1} x=\sin \cot ^{-1} \tan \theta \\ =\sin \cot ^{-1}(\frac{\sqrt{1-x^{2}}}{x}) \ \text{put x=} \sin \theta\\ =\sin \cot ^{-1}(\frac{\sqrt{1-\sin ^{2} \theta}}{\sin \theta}) \\ =\sin \cot ^{-1}(\cot \theta)\\ =\sin \theta\\ =\sqrt{1-x^2}

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jitender.kumar

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