Question

if we have a trigonometric expression as $[{{\cot }^{-1}}x]+[{{\cos }^{-1}}x]=0$ , where [ . ] denotes greatest integer function, then the complete set of values of x is
( a ) (cos1, 1]
( b ) ( cos1, - cos1)
( c ) (cot1, 1]
( d ) none of these

Answer 1

What we will do is we will first draw the graph of ${{\cos }^{-1}}x$ and ${{\cot }^{-1}}x$ using concept of greatest integer function. Then we will first see for what values of x, ${{\cos }^{-1}}x$ and ${{\cot }^{-1}}x$ is equals to zero. Then we will take intersection of set of domain of values of x for $[{{\cos }^{-1}}x]=0$ and $[{{\cot }^{-1}}x]=0$.

Complete step-by-step solution:
Before we start the question, let us see what is the greatest integer function and what are its properties.
Function y = [ x ] is called greatest integer function which means the greatest integer less than or equals to x. also, if n belongs to set of integer, then y = [ x ] = n if $n\le xFor example if we put x = -3.1 in y = [ x ], then y = - 4 and if we put x = 0.2 in y = [ x ], then y = 0. Graph of y = [ x ] is given as,  Graph of$y=[{{\cot }^{-1}}x]$is given as,  Graph of$y=[{{\cos }^{-1}}x]$is given as  Now, in question it is given that$[{{\cot }^{-1}}x]+[{{\cos }^{-1}}x]=0$Now, we know that range of${{\cos }^{-1}}x$is$0\le {{\cos }^{-1}}x\le \pi $. So from seeing the range of${{\cos }^{-1}}x$, we can say that${{\cos }^{-1}}x$is always positive for all values of x. where x belongs to a set of real numbers. Also, we know that range of${{\cot }^{-1}}x$is$0<{{\cot }^{-1}}x\le \pi $. So from seeing the range of${{\cot }^{-1}}x$, we can say that${{\cos }^{-1}}x$is always positive for all values of x. where x belongs to a set of real numbers. Now for, values of${{\cot }^{-1}}x$and${{\cos }^{-1}}x$,$[{{\cot }^{-1}}x]+[{{\cos }^{-1}}x]=0$is true when both$[{{\cot }^{-1}}x]$and$[{{\cos }^{-1}}x]$are equals to zero. Now, from graph of$[{{\cos }^{-1}}x]$,$[{{\cos }^{-1}}x]$is always zero for values between cos1 and 1 that is,$[{{\cos }^{-1}}x]=0;x\in (\cos 1,1]$….( i ) Now, from graph of$[{{\cot }^{-1}}x]$,$[{{\cot }^{-1}}x]$is always zero for values between cot1 and$\infty $that is,$[{{\cot }^{-1}}x]=0;x\in (\cot 1,\infty )$……( ii ) Taking intersection of equation ( i ) and ( ii ) So, set of all values x for which$[{{\cot }^{-1}}x]+[{{\cos }^{-1}}x]=0$is$x\in (\cot 1,1]$
Hence, option ( c ) is true.

Note: For finding domain of functions which are formed by combination of some function of x and greatest integer function, knowledge of graph is must. If we have $f(x)={{f}_{1}}(x)+{{f}_{2}}(x)$ , then domain of f(x) is equals to domain of ${{f}_{1}}(x)\cap {{f}_{2}}(x)$.