   Question Answers

# If $\sin \left[ {{{\cot }^{ - 1}}\left( {x + 1} \right)} \right] = \cos \left( {{{\tan }^{ - 1}}x} \right)$, then find the value of $x$.  Hint: In this question, we will proceed by converting ${\cot ^{ - 1}}$ in terms of ${\sin ^{ - 1}}$ and ${\tan ^{ - 1}}$ in terms of ${\cos ^{ - 1}}$ by using the formula ${\cot ^{ - 1}}\left( {x + 1} \right) = {\sin ^{ - 1}}\dfrac{1}{{\sqrt {1 + {{\left( {x + 1} \right)}^2}} }}$ and ${\tan ^{ - 1}}x = {\cos ^{ - 1}}\dfrac{1}{{\sqrt {1 + {x^2}} }}$. So, use this concept to reach the solution of the problem.

Given that $\sin \left[ {{{\cot }^{ - 1}}\left( {x + 1} \right)} \right] = \cos \left( {{{\tan }^{ - 1}}x} \right)$
By using the formula ${\cot ^{ - 1}}\left( {x + 1} \right) = {\sin ^{ - 1}}\dfrac{1}{{\sqrt {1 + {{\left( {x + 1} \right)}^2}} }}$ and ${\tan ^{ - 1}}x = {\cos ^{ - 1}}\dfrac{1}{{\sqrt {1 + {x^2}} }}$
$\Rightarrow \sin \left[ {{{\sin }^{ - 1}}\dfrac{1}{{\sqrt {1 + {{\left( {1 + x} \right)}^2}} }}} \right] = \cos \left[ {{{\cos }^{ - 1}}\dfrac{1}{{\sqrt {1 + {x^2}} }}} \right]$
We know that, $\sin \left( {{{\sin }^{ - 1}}A} \right) = A$ and $\cos \left( {{{\cos }^{ - 1}}A} \right) = A$
$\Rightarrow \dfrac{1}{{\sqrt {1 + {{\left( {1 + x} \right)}^2}} }} = \dfrac{1}{{\sqrt {1 + {x^2}} }} \\ \Rightarrow \sqrt {1 + {x^2}} = \sqrt {1 + {{\left( {1 + x} \right)}^2}} \\$
Squaring on both sides, we have
$\Rightarrow 1 + {x^2} = 1 + {\left( {1 + x} \right)^2} \\ \Rightarrow 1 + {x^2} = 1 + 1 + 2x + {x^2} \\ \Rightarrow 1 + {x^2} = 2 + 2x + {x^2} \\ \Rightarrow 1 + {x^2} - 2 - {x^2} = 2x \\ \Rightarrow - 1 = 2x \\ \therefore x = - \dfrac{1}{2} \\$

Thus, the value of $x$ is $- \dfrac{1}{2}$

Note: To solve these kinds of questions, students must be familiar with all the formulae in trigonometry and inverse trigonometry. If we didn’t remember the formulae we can draw the corresponding right angle triangle then we convert the terms of ${\cot ^{ - 1}}$ and ${\tan ^{ - 1}}$ in terms of ${\sin ^{ - 1}}$ and ${\cos ^{ - 1}}$ respectively. But it consumes a lot of time. So, do remember the formulae in order to solve them easily.
Difference Between Left and Right Ventricle  Difference Between Left Kidney and Right Kidney  CBSE Class 12 Maths Chapter-2 Inverse Trigonometric Functions Formula  Inverse Trigonometric Functions  Sin Cos Formula  Right Angled Triangle Constructions  Trigonometric Functions  CBSE Class 12 Maths Chapter-1 Relations and Functions Formula  Cos 360  Sin 120  Important Questions for CBSE Class 12 Maths Chapter 2 - Inverse Trigonometric Functions  Important Questions for CBSE Class 12 Maths Chapter 1 - Relations and Functions  NCERT Books Free Download for Class 12 Maths Chapter-2 Inverse Trigonometric Functions  NCERT Books Free Download for Class 12 Maths Chapter-1 Relations and Functions  Important Questions for CBSE Class 11 Maths Chapter 3 - Trigonometric Functions  Important Questions for CBSE Class 12 Biology Chapter 1 - Reproduction in Organism  Important Questions for CBSE Class 12 Physics Chapter 1 - Electric Charges and Fields  Important Questions for CBSE Class 12 Chemistry Chapter 1 - The Solid State  Important Questions for CBSE Class 12 Macro Economics Chapter 1 - Introduction to Macro Economics  Important Questions for CBSE Class 12 Hindi Antral Chapter 1 - Surdas Ki Jhopdi  Maths Question Paper for CBSE Class 12 - 2016 Set 1 E  Maths Question Paper for CBSE Class 12 - 2016 Set 1 S  Maths Question Paper for CBSE Class 12 - 2016 Set 1 C  Maths Question Paper for CBSE Class 12 - 2016 Set 1 N  CBSE Class 12 Maths Question Paper 2020  Maths Question Paper for CBSE Class 12 - 2013  Chemistry Question Paper for CBSE Class 12 - 2016 Set 1 E  Chemistry Question Paper for CBSE Class 12 - 2016 Set 1 S  Chemistry Question Paper for CBSE Class 12 - 2016 Set 1 N  Chemistry Question Paper for CBSE Class 12 - 2016 Set 1 C  NCERT Solutions for Class 12 Maths Chapter 2  NCERT Class 12 Maths Chapter 1  RD Sharma Class 12 Maths Solutions Chapter 4 - Inverse Trigonometric Functions  RD Sharma Class 12 Solutions Chapter 1 - Relations (Ex 1.1) Exercise 1.1  NCERT Solutions for Class 12 Maths Chapter 1 Exercise 1.1 (Ex 1.1)  RD Sharma Solutions for Class 9 Maths Chapter 20 - Surface Area and Volume of A Right Circular Cone  RD Sharma Solutions for Class 9 Maths Chapter 19 - Surface Area and Volume of A Right Circular Cylinder  RD Sharma Class 12 Solutions Chapter 1 - Relations (Ex 1.2) Exercise 1.2  NCERT Solutions for Class 12 Maths Chapter 2 Inverse Trigonometric Functions In Hindi  RS Aggarwal Class 12 Solutions Chapter-4 Inverse Trigonometric Functions  