Answer
Verified
395.7k+ views
Hint: We know that ${{\cot }^{-1}}x+{{\tan }^{-1}}x=\dfrac{\pi }{2}$ hence we get x = \[\dfrac{\pi }{2}\]. Now since we know the value of x. we can easily find the value of sinx.
Complete step by step answer:
Now we are given that ${{\cot }^{-1}}[{{\left( \cos \alpha \right)}^{\dfrac{1}{2}}}]+{{\tan }^{-1}}[{{\left( \cos \alpha \right)}^{\dfrac{1}{2}}}]=x$ .
Let us say ${{\left( \cos \alpha \right)}^{\dfrac{1}{2}}}$ is equal to t.
Now we will note that the range of ${{\cos }^{-1}}\theta $ is $[0,\pi ]$ and hence the output of $\cos \alpha $ is also real
Hence we have $t={{(\cos \alpha )}^{\dfrac{1}{2}}}$ is a real number.
Substituting this in the above equation we get
${{\cot }^{-1}}t+{{\tan }^{-1}}t=x.................(1)$
Now we know that for all real numbers x the identity ${{\cot }^{-1}}x+{{\tan }^{-1}}x=\dfrac{\pi }{2}$ is true.
Now since this identity is true for all real numbers x and does not depend on x this identity is also true for t which is a real number.
Hence applying this to equation (1) we get $x=\dfrac{\pi }{2}$
Hence now we have the value of x is $\dfrac{\pi }{2}$.
Now we have to find the value of $\sin x$
Now since $x=\dfrac{\pi }{2}$ we have $\sin x=\sin \dfrac{\pi }{2}$.
We know that the value of $\sin \dfrac{\pi }{2}$ is 1.
Hence we get the value of $\sin x$ is equal to 1.
So, the correct answer is “Option A”.
Note: Here the equation is given in a confusing format which is nothing but a equation in form of ${{\cot }^{-1}}a+{{\tan }^{-1}}a$ and for that we know the identity = ${{\cot }^{-1}}x+{{\tan }^{-1}}x=\dfrac{\pi }{2}$ we can directly apply it to solve our question.
Complete step by step answer:
Now we are given that ${{\cot }^{-1}}[{{\left( \cos \alpha \right)}^{\dfrac{1}{2}}}]+{{\tan }^{-1}}[{{\left( \cos \alpha \right)}^{\dfrac{1}{2}}}]=x$ .
Let us say ${{\left( \cos \alpha \right)}^{\dfrac{1}{2}}}$ is equal to t.
Now we will note that the range of ${{\cos }^{-1}}\theta $ is $[0,\pi ]$ and hence the output of $\cos \alpha $ is also real
Hence we have $t={{(\cos \alpha )}^{\dfrac{1}{2}}}$ is a real number.
Substituting this in the above equation we get
${{\cot }^{-1}}t+{{\tan }^{-1}}t=x.................(1)$
Now we know that for all real numbers x the identity ${{\cot }^{-1}}x+{{\tan }^{-1}}x=\dfrac{\pi }{2}$ is true.
Now since this identity is true for all real numbers x and does not depend on x this identity is also true for t which is a real number.
Hence applying this to equation (1) we get $x=\dfrac{\pi }{2}$
Hence now we have the value of x is $\dfrac{\pi }{2}$.
Now we have to find the value of $\sin x$
Now since $x=\dfrac{\pi }{2}$ we have $\sin x=\sin \dfrac{\pi }{2}$.
We know that the value of $\sin \dfrac{\pi }{2}$ is 1.
Hence we get the value of $\sin x$ is equal to 1.
So, the correct answer is “Option A”.
Note: Here the equation is given in a confusing format which is nothing but a equation in form of ${{\cot }^{-1}}a+{{\tan }^{-1}}a$ and for that we know the identity = ${{\cot }^{-1}}x+{{\tan }^{-1}}x=\dfrac{\pi }{2}$ we can directly apply it to solve our question.
Recently Updated Pages
Basicity of sulphurous acid and sulphuric acid are
Assertion The resistivity of a semiconductor increases class 13 physics CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
What is the stopping potential when the metal with class 12 physics JEE_Main
The momentum of a photon is 2 times 10 16gm cmsec Its class 12 physics JEE_Main
Using the following information to help you answer class 12 chemistry CBSE
Trending doubts
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
Two charges are placed at a certain distance apart class 12 physics CBSE
Difference Between Plant Cell and Animal Cell
What organs are located on the left side of your body class 11 biology CBSE
Change the following sentences into negative and interrogative class 10 english CBSE
The planet nearest to earth is A Mercury B Venus C class 6 social science CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
What is BLO What is the full form of BLO class 8 social science CBSE