
If $A + B = \dfrac{\pi }{4}$ then $\left( {1 + \tan A} \right)\left( {1 + \tan B} \right)$ is equal to
A. $1$
B. $2$
C. $3$
D. None of these
Answer
162.3k+ views
Hint: In order to solve this type of question, first we will consider the given equation and solve it by taking tan on both the sides. Then, we will apply a suitable trigonometric identity(tangent sum) to it and solve it further in order to get the desired answer.
Formula used:
$\left[ {\because \tan \dfrac{\pi }{4} = 1} \right]$
$\left[ {\because \tan \left( {A + B} \right) = \dfrac{{\tan A + \tan B}}{{1 - \tan A\tan B}}} \right]$
Complete step by step solution:
We are given that,
$A + B = \dfrac{\pi }{4}$
Taking tan on both sides
$\tan \left( {A + B} \right) = \tan \dfrac{\pi }{4}$
$\tan \left( {A + B} \right) = 1$ $\left[ {\because \tan \dfrac{\pi }{4} = 1} \right]$
Solving it,
$\dfrac{{\tan A + \tan B}}{{1 - \tan A\tan B}} = 1$ $\left[ {\because \tan \left( {A + B} \right) = \dfrac{{\tan A + \tan B}}{{1 - \tan A\tan B}}} \right]$
\[\tan A + \tan B = 1\left( {1 - \tan A\tan B} \right)\]
Add $\left( {1 + \tan A\tan B} \right)$ to both sides,
\[\tan A + 1 + \tan B + \tan A\tan B = 2\]
Taking common,
\[1\left( {1 + \tan A} \right) + \tan B\left( {1 + \tan A} \right) = 2\]
\[\left( {1 + \tan A} \right)\left( {1 + \tan B} \right) = 2\]
$\therefore $ The correct option is B.
Note: The key concept of solving this type of question is to be careful with the simplification part. Choose the suitable trigonometric identities and be very sure while simplifying them. This type of question requires the use of correct application of trigonometric rules to get the correct answer.
Formula used:
$\left[ {\because \tan \dfrac{\pi }{4} = 1} \right]$
$\left[ {\because \tan \left( {A + B} \right) = \dfrac{{\tan A + \tan B}}{{1 - \tan A\tan B}}} \right]$
Complete step by step solution:
We are given that,
$A + B = \dfrac{\pi }{4}$
Taking tan on both sides
$\tan \left( {A + B} \right) = \tan \dfrac{\pi }{4}$
$\tan \left( {A + B} \right) = 1$ $\left[ {\because \tan \dfrac{\pi }{4} = 1} \right]$
Solving it,
$\dfrac{{\tan A + \tan B}}{{1 - \tan A\tan B}} = 1$ $\left[ {\because \tan \left( {A + B} \right) = \dfrac{{\tan A + \tan B}}{{1 - \tan A\tan B}}} \right]$
\[\tan A + \tan B = 1\left( {1 - \tan A\tan B} \right)\]
Add $\left( {1 + \tan A\tan B} \right)$ to both sides,
\[\tan A + 1 + \tan B + \tan A\tan B = 2\]
Taking common,
\[1\left( {1 + \tan A} \right) + \tan B\left( {1 + \tan A} \right) = 2\]
\[\left( {1 + \tan A} \right)\left( {1 + \tan B} \right) = 2\]
$\therefore $ The correct option is B.
Note: The key concept of solving this type of question is to be careful with the simplification part. Choose the suitable trigonometric identities and be very sure while simplifying them. This type of question requires the use of correct application of trigonometric rules to get the correct answer.
Recently Updated Pages
How To Find Mean Deviation For Ungrouped Data

Difference Between Molecule and Compound: JEE Main 2024

Ammonium Hydroxide Formula - Chemical, Molecular Formula and Uses

Difference Between Area and Surface Area: JEE Main 2024

Difference Between Work and Power: JEE Main 2024

Difference Between Acetic Acid and Glacial Acetic Acid: JEE Main 2024

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

Displacement-Time Graph and Velocity-Time Graph for JEE

Degree of Dissociation and Its Formula With Solved Example for JEE

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

JoSAA JEE Main & Advanced 2025 Counselling: Registration Dates, Documents, Fees, Seat Allotment & Cut‑offs

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

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 Chapter 6 Permutations and Combinations

NCERT Solutions for Class 11 Maths In Hindi Chapter 1 Sets

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