solution of question asked by Mr. Ravi Raj Anand -II

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Ques: If [{\cot ^{ - 1}}x] + [{\cos ^{ - 1}}x] = 0 then find the value of x, where [.] denotes the greatest integer function.

 Sol.
Since [.] gives the integer value and both the functions i.e. {\cot ^{ - 1}}x and {\cos ^{ - 1}}x are always greater than '0'.

 Therefore, sum of [{\cot ^{ - 1}}x] + [{\cos ^{ - 1}}x] = 0 when BOTH are '0'.

 STEPS : We separately find the interval of x for both the expression i.e. {\cot ^{ - 1}}x and {\cos ^{ - 1}}x  for which they are zero and then take the INTERSECTION of both the intervals.
1. First consider [{\cos ^{ - 1}}x] ,

 see the graph,

Clearly, from the graph,
[{\cos ^{ - 1}}x] = 0 , if \cos 1 < x \le 1 or x \in (\cos 1,1]  ..................... (1)

 2. Consider, [{\cot ^{ - 1}}x]

 understand the below graph,

 Clearly from the graph,
[{\cot ^{ - 1}}x] = 0 , if x > \cot 1 or x \in (\cot 1,\infty ) .................... (2)

 Now, we have to take the intersection of (1) and (2) i.e. to find the values of x for which both [{\cos ^{ - 1}}x] and
[{\cot ^{ - 1}}x] are '0'.

 so, x \in (\cot 1,\infty )\bigcap {(\cos 1,1]}

But to take intersection of these two intervals we have to find where these numbers viz. cos1, cot1 and 1 lie on number line.

 since, \cot 1 = \frac{{\cos 1}}{{\sin 1}} and we know that sin1 < 1
therefore, cot1 > cos1

 we know that from above graph cos1 < 1.

 consider the below graph of cot x,
when x = pi/4 , cot x = 1 , and if x = 1 , cot x = cot 1,
clearly cot 1 < 1
x \in (\cot 1,\infty )\bigcap {(\cos 1,1]}

therefore, x \in (\cot 1,1]

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