
If \[cosec\left[ {{{\tan }^{ - 1}}\left( {\dfrac{1}{7}} \right) + {{\cot }^{ - 1}}\left( x \right)} \right] = 1\], find the value of \[x\].
Answer
411.3k+ views
Hint: Here, in the question, we have been given a trigonometric equation including an unknown variable and we are asked to find the value of that variable. On the right hand side of the equation, \[1\] is given and on the left hand side, function of cosecant is given. We will put the cosecant function on the right hand such that the angle of cosecant gives the value \[1\]. And then, we will simplify using inverse trigonometric identity and get the desired result.
Formula used:
\[{\tan ^{ - 1}}\dfrac{1}{x} = {\cot ^{ - 1}}x \\
{\cot ^{ - 1}}\left( { - x} \right) = \pi - {\cot ^{ - 1}}x \\
\cot \left( {{{\cot }^{ - 1}}x} \right) = x \\
\cot \left( {x + y} \right) = \dfrac{{\cot x\cot y - 1}}{{\cot y + \cot x}} \]
Complete step-by-step solution:
Given equation: \[cosec\left[ {{{\tan }^{ - 1}}\left( {\dfrac{1}{7}} \right) + {{\cot }^{ - 1}}\left( x \right)} \right] = 1\]
Now, we have a cosecant function on the left hand side. To make the equation simpler, we will have to bring the cosecant function on the right hand side. For that, we will put the cosecant function such that the angle of the cosecant results in the value \[1\]. And we know, at \[\dfrac{\pi }{2}\] cosecant function is equal to \[1\].
Therefore, the equation becomes as,
\[
cosec\left[ {{{\tan }^{ - 1}}\left( {\dfrac{1}{7}} \right) + {{\cot }^{ - 1}}\left( x \right)} \right] = cosec\left( {\dfrac{\pi }{2}} \right) \\
\Rightarrow {\tan ^{ - 1}}\left( {\dfrac{1}{7}} \right) + {\cot ^{ - 1}}\left( x \right) = \dfrac{\pi }{2} \\
\]
Using the identity, \[{\tan ^{ - 1}}\dfrac{1}{x} = {\cot ^{ - 1}}x\], we get,
\[ {\cot ^{ - 1}}\left( 7 \right) + {\cot ^{ - 1}}\left( x \right) = \dfrac{\pi }{2} \\
\Rightarrow {\cot ^{ - 1}}\left( 7 \right) - \dfrac{\pi }{2} = - {\cot ^{ - 1}}\left( x \right) \]
Using identity, \[\left[\because {\cot ^{ - 1}}\left( { - x} \right) = \pi - {\cot ^{ - 1}}x
\Rightarrow - {\cot ^{ - 1}}\left( x \right) = {\cot ^{ - 1}}\left( { - x} \right) - \pi \right]\] ,
we get,
\[{\cot ^{ - 1}}\left( 7 \right) - \dfrac{\pi }{2} = {\cot ^{ - 1}}\left( { - x} \right) - \pi \\ \]
Adding \[\pi \] both sides, we get,
\[{\cot ^{ - 1}}\left( 7 \right) + \dfrac{\pi }{2} = {\cot ^{ - 1}}\left( { - x} \right)\]
Taking \[\cot \] both sides, we get,
\[\cot \left( {{{\cot }^{ - 1}}\left( 7 \right) + \dfrac{\pi }{2}} \right) = \cot \left( {{{\cot }^{ - 1}}\left( { - x} \right)} \right) \\
\Rightarrow \cot \left( {{{\cot }^{ - 1}}\left( 7 \right) + \dfrac{\pi }{2}} \right) = - x \\
\] \[\left[ {\because \cot \left( {{{\cot }^{ - 1}}x} \right) = x} \right]\]
Using identity \[\cot \left( {x + y} \right) = \dfrac{{\cot x\cot y - 1}}{{\cot y + \cot x}}\], we obtain,
\[\dfrac{{\cot \left( {{{\cot }^{ - 1}}7} \right)\cot \left( {\dfrac{\pi }{2}} \right) - 1}}{{\cot \left( {\dfrac{\pi }{2}} \right) + \cot \left( {{{\cot }^{ - 1}}7} \right)}} = - x\]
Now, we know that, \[\cot \dfrac{\pi }{2} = 0\], then,
\[\dfrac{{ - 1}}{7} = - x\]
Multiplying by \[\left( { - 1} \right)\] both sides, we get,
\[x = \dfrac{1}{7}\]
Hence, the value of \[x\] is \[\dfrac{1}{7}\].
Note: The symbol \[{\tan ^{ - 1}}x\] should not be confused with \[{\left( {\tan x} \right)^{ - 1}}\]. In-fact \[{\tan ^{ - 1}}x\] is an angle, the value of whose tangent is \[x\]. The only key concept to solve such types of questions is that we must remember all the basic trigonometric identities and their application. This will help us to solve almost all the questions.
Formula used:
\[{\tan ^{ - 1}}\dfrac{1}{x} = {\cot ^{ - 1}}x \\
{\cot ^{ - 1}}\left( { - x} \right) = \pi - {\cot ^{ - 1}}x \\
\cot \left( {{{\cot }^{ - 1}}x} \right) = x \\
\cot \left( {x + y} \right) = \dfrac{{\cot x\cot y - 1}}{{\cot y + \cot x}} \]
Complete step-by-step solution:
Given equation: \[cosec\left[ {{{\tan }^{ - 1}}\left( {\dfrac{1}{7}} \right) + {{\cot }^{ - 1}}\left( x \right)} \right] = 1\]
Now, we have a cosecant function on the left hand side. To make the equation simpler, we will have to bring the cosecant function on the right hand side. For that, we will put the cosecant function such that the angle of the cosecant results in the value \[1\]. And we know, at \[\dfrac{\pi }{2}\] cosecant function is equal to \[1\].
Therefore, the equation becomes as,
\[
cosec\left[ {{{\tan }^{ - 1}}\left( {\dfrac{1}{7}} \right) + {{\cot }^{ - 1}}\left( x \right)} \right] = cosec\left( {\dfrac{\pi }{2}} \right) \\
\Rightarrow {\tan ^{ - 1}}\left( {\dfrac{1}{7}} \right) + {\cot ^{ - 1}}\left( x \right) = \dfrac{\pi }{2} \\
\]
Using the identity, \[{\tan ^{ - 1}}\dfrac{1}{x} = {\cot ^{ - 1}}x\], we get,
\[ {\cot ^{ - 1}}\left( 7 \right) + {\cot ^{ - 1}}\left( x \right) = \dfrac{\pi }{2} \\
\Rightarrow {\cot ^{ - 1}}\left( 7 \right) - \dfrac{\pi }{2} = - {\cot ^{ - 1}}\left( x \right) \]
Using identity, \[\left[\because {\cot ^{ - 1}}\left( { - x} \right) = \pi - {\cot ^{ - 1}}x
\Rightarrow - {\cot ^{ - 1}}\left( x \right) = {\cot ^{ - 1}}\left( { - x} \right) - \pi \right]\] ,
we get,
\[{\cot ^{ - 1}}\left( 7 \right) - \dfrac{\pi }{2} = {\cot ^{ - 1}}\left( { - x} \right) - \pi \\ \]
Adding \[\pi \] both sides, we get,
\[{\cot ^{ - 1}}\left( 7 \right) + \dfrac{\pi }{2} = {\cot ^{ - 1}}\left( { - x} \right)\]
Taking \[\cot \] both sides, we get,
\[\cot \left( {{{\cot }^{ - 1}}\left( 7 \right) + \dfrac{\pi }{2}} \right) = \cot \left( {{{\cot }^{ - 1}}\left( { - x} \right)} \right) \\
\Rightarrow \cot \left( {{{\cot }^{ - 1}}\left( 7 \right) + \dfrac{\pi }{2}} \right) = - x \\
\] \[\left[ {\because \cot \left( {{{\cot }^{ - 1}}x} \right) = x} \right]\]
Using identity \[\cot \left( {x + y} \right) = \dfrac{{\cot x\cot y - 1}}{{\cot y + \cot x}}\], we obtain,
\[\dfrac{{\cot \left( {{{\cot }^{ - 1}}7} \right)\cot \left( {\dfrac{\pi }{2}} \right) - 1}}{{\cot \left( {\dfrac{\pi }{2}} \right) + \cot \left( {{{\cot }^{ - 1}}7} \right)}} = - x\]
Now, we know that, \[\cot \dfrac{\pi }{2} = 0\], then,
\[\dfrac{{ - 1}}{7} = - x\]
Multiplying by \[\left( { - 1} \right)\] both sides, we get,
\[x = \dfrac{1}{7}\]
Hence, the value of \[x\] is \[\dfrac{1}{7}\].
Note: The symbol \[{\tan ^{ - 1}}x\] should not be confused with \[{\left( {\tan x} \right)^{ - 1}}\]. In-fact \[{\tan ^{ - 1}}x\] is an angle, the value of whose tangent is \[x\]. The only key concept to solve such types of questions is that we must remember all the basic trigonometric identities and their application. This will help us to solve almost all the questions.
Recently Updated Pages
Master Class 12 Economics: Engaging Questions & Answers for Success

Master Class 12 Maths: Engaging Questions & Answers for Success

Master Class 12 Biology: Engaging Questions & Answers for Success

Master Class 12 Physics: Engaging Questions & Answers for Success

Master Class 12 Business Studies: Engaging Questions & Answers for Success

Master Class 12 English: Engaging Questions & Answers for Success

Trending doubts
Which are the Top 10 Largest Countries of the World?

Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE

Draw a labelled sketch of the human eye class 12 physics CBSE

What is the Full Form of PVC, PET, HDPE, LDPE, PP and PS ?

What is a transformer Explain the principle construction class 12 physics CBSE

What are the major means of transport Explain each class 12 social science CBSE
