
If in a triangle ABC, angle C is \[45^\circ \], then \[\left( {1 + cotA} \right)\left( {1 + cotB} \right)\]=
A. \[ - 1\]
B. \[2\]
C. \[3\]
D. \[\sqrt {\frac{1}{2}} \]
Answer
161.4k+ views
Hint
To solve this problem, we need to use the law of sines. The law of sines states that in a right triangle, the sum of the two angles is equal to\[120^\circ \]. So, \[\left( {1 + cotA} \right)\left( {1 + cotB} \right) = 120^\circ \].
According to the law of sine, or sine law, the ratio of a triangle's side length to the sine of the opposing angle remains constant for all three sides. The sine rule is another name for it. The relationship between the sides and angles of non-right (oblique) triangles is known as the Law of Sines. It simply asserts that for all sides and angles of a given triangle, the ratio of the length of a side to the sine of the angle opposite that side is the same.
Complete step-by-step solution:
The given equation is\[\angle C = 45^\circ \]and
\[A + B = 180^\circ - 45^\circ = 135^\circ \]
\[\cot (A + B) = \frac{{\cot A \cot B - 1}}{{\cot A + \cot B}}\]
As the value of A+B is already known, the equation becomes,
\[\cot (135^\circ ) = - 1 = \frac{{\cot A \cot B - 1}}{{\cot A + \cot B}}\]
\[\cot A + \cot B = 1 - \cot A\cot B\]
\[1 + \cot A + \cot B + \cot A\cot B\]
Then, the answer is equal to \[2\]
\[(1 + \cot A)(1 + \cot B) = 2\]
So, the option B is correct.
Note
Use an ordering of operations to solve the triangle issue. The parenthesis is used to emphasize that the contents should be considered first before anything else. Since angle C is \[45\] degrees in this instance, it is analyzed first, leading to the results: \[\left( {1 + cotA} \right)\left( {1 + cotB} \right) = 2{\rm{ }}or{\rm{ }}1 + \left( {cot\left( {45} \right)} \right) = \left( {1 + cotB} \right)\].
A phrase, word, or sentence added to writing as supplemental information or an afterthought is known as a parenthesis. It is emphasized with bracketed text, commas, or dashes.
Multiplying numbers is the second application of parentheses in mathematics. When parenthesis is used in an equation and there is no arithmetic operation, multiplication must be used.
To solve this problem, we need to use the law of sines. The law of sines states that in a right triangle, the sum of the two angles is equal to\[120^\circ \]. So, \[\left( {1 + cotA} \right)\left( {1 + cotB} \right) = 120^\circ \].
According to the law of sine, or sine law, the ratio of a triangle's side length to the sine of the opposing angle remains constant for all three sides. The sine rule is another name for it. The relationship between the sides and angles of non-right (oblique) triangles is known as the Law of Sines. It simply asserts that for all sides and angles of a given triangle, the ratio of the length of a side to the sine of the angle opposite that side is the same.
Complete step-by-step solution:
The given equation is\[\angle C = 45^\circ \]and
\[A + B = 180^\circ - 45^\circ = 135^\circ \]
\[\cot (A + B) = \frac{{\cot A \cot B - 1}}{{\cot A + \cot B}}\]
As the value of A+B is already known, the equation becomes,
\[\cot (135^\circ ) = - 1 = \frac{{\cot A \cot B - 1}}{{\cot A + \cot B}}\]
\[\cot A + \cot B = 1 - \cot A\cot B\]
\[1 + \cot A + \cot B + \cot A\cot B\]
Then, the answer is equal to \[2\]
\[(1 + \cot A)(1 + \cot B) = 2\]
So, the option B is correct.
Note
Use an ordering of operations to solve the triangle issue. The parenthesis is used to emphasize that the contents should be considered first before anything else. Since angle C is \[45\] degrees in this instance, it is analyzed first, leading to the results: \[\left( {1 + cotA} \right)\left( {1 + cotB} \right) = 2{\rm{ }}or{\rm{ }}1 + \left( {cot\left( {45} \right)} \right) = \left( {1 + cotB} \right)\].
A phrase, word, or sentence added to writing as supplemental information or an afterthought is known as a parenthesis. It is emphasized with bracketed text, commas, or dashes.
Multiplying numbers is the second application of parentheses in mathematics. When parenthesis is used in an equation and there is no arithmetic operation, multiplication must be used.
Recently Updated Pages
If there are 25 railway stations on a railway line class 11 maths JEE_Main

Minimum area of the circle which touches the parabolas class 11 maths JEE_Main

Which of the following is the empty set A x x is a class 11 maths JEE_Main

The number of ways of selecting two squares on chessboard class 11 maths JEE_Main

Find the points common to the hyperbola 25x2 9y2 2-class-11-maths-JEE_Main

A box contains 6 balls which may be all of different class 11 maths JEE_Main

Trending doubts
JEE Main 2025 Session 2: Application Form (Out), Exam Dates (Released), Eligibility, & More

JEE Main 2025: Derivation of Equation of Trajectory in Physics

Electric Field Due to Uniformly Charged Ring for JEE Main 2025 - Formula and Derivation

Displacement-Time Graph and Velocity-Time Graph for JEE

Degree of Dissociation and Its Formula With Solved Example for JEE

Free Radical Substitution Mechanism of Alkanes for JEE Main 2025

Other Pages
JEE Advanced Marks vs Ranks 2025: Understanding Category-wise Qualifying Marks and Previous Year Cut-offs

JEE Advanced 2025: Dates, Registration, Syllabus, Eligibility Criteria and More

JEE Advanced Weightage 2025 Chapter-Wise for Physics, Maths and Chemistry

NCERT Solutions for Class 11 Maths Chapter 4 Complex Numbers and Quadratic Equations

NCERT Solutions for Class 11 Maths In Hindi Chapter 1 Sets

NCERT Solutions for Class 11 Maths Chapter 6 Permutations and Combinations
