
If \[\cos (A - B) = \dfrac{3}{5}\] and \[\tan A\tan B = 2\] , then choose the correct option.
A. \[\cos A\cos B = \dfrac{1}{5}\]
B.\[\sin A\sin B = - \dfrac{2}{5}\]
C.\[\cos (A + B) = - \dfrac{1}{5}\]
D.\[\sin A\cos B = \dfrac{4}{5}\]
Answer
162.9k+ views
Hint: Write \[\tan A\tan B\] in terms of \[\sin A\],\[\sin B\],\[\cos A\] and \[\cos B\].From there we will get the relation in \[\sin A\sin B\] and \[\cos A\cos B\]. Expand the formulae of \[\cos (A - B)\] and use the relations obtained earlier to get the desired values.
Formula Used
1.\[\tan \theta = \dfrac{{\sin \theta }}{{\cos \theta }}\]
2.\[\cos (A - B) = \cos A\cos B + \sin A\sin B\]
3. \[\cos (A + B) = \cos A\cos B - \sin A\sin B\]
Complete step by step solution
Given- \[\tan A\tan B = 2\]
We know that \[\tan \theta = \dfrac{{\sin \theta }}{{\cos \theta }}\] ,using this identity in above relation
\[\dfrac{{\sin A\sin B}}{{\cos A\cos B}} = 2\]
Cross-multiplying the terms
\[\sin A\sin B = 2\cos A\cos B\] (1)
Again, \[\cos (A - B) = \dfrac{3}{5}\] (Given)
Applying the formula \[\cos (A - B) = \cos A\cos B + \sin A\sin B\]
\[\cos A\cos B + \sin A\sin B = \dfrac{3}{5}\]
Using equation (1)
\[\cos A\cos B + 2\cos A\cos B = \dfrac{3}{5}\]
\[ \Rightarrow 3\cos A\cos B = \dfrac{3}{5}\]
\[ \Rightarrow \cos A\cos B = \dfrac{1}{5}\]
Again, using equation (1)
\[\sin A\sin B = 2\cos A\cos B\]
\[ \Rightarrow \sin A\sin B = 2*\dfrac{1}{5}\]
\[ \Rightarrow \sin A\sin B = \dfrac{2}{5}\]
Now using the formula \[\cos (A + B) = \cos A\cos B - \sin A\sin B\]
\[\cos (A + B) = \dfrac{1}{5} - \dfrac{2}{5}\]
\[ \Rightarrow \cos (A + B) = - \dfrac{1}{5}\]
Hence option C is the correct answer.
Note: The values \[\sin A\cos B\] and \[\cos A\sin B\] are not given. Thus we will give equations to find the values of \[\sin A\cos B\] and \[\cos A\sin B\].
Students must not get confused in the sum and difference formulas of cosine; they are as follows –
\[\cos (A - B) = \cos A\cos B + \sin A\sin B\]
\[\cos (A + B) = \cos A\cos B - \sin A\sin B\]
Formula Used
1.\[\tan \theta = \dfrac{{\sin \theta }}{{\cos \theta }}\]
2.\[\cos (A - B) = \cos A\cos B + \sin A\sin B\]
3. \[\cos (A + B) = \cos A\cos B - \sin A\sin B\]
Complete step by step solution
Given- \[\tan A\tan B = 2\]
We know that \[\tan \theta = \dfrac{{\sin \theta }}{{\cos \theta }}\] ,using this identity in above relation
\[\dfrac{{\sin A\sin B}}{{\cos A\cos B}} = 2\]
Cross-multiplying the terms
\[\sin A\sin B = 2\cos A\cos B\] (1)
Again, \[\cos (A - B) = \dfrac{3}{5}\] (Given)
Applying the formula \[\cos (A - B) = \cos A\cos B + \sin A\sin B\]
\[\cos A\cos B + \sin A\sin B = \dfrac{3}{5}\]
Using equation (1)
\[\cos A\cos B + 2\cos A\cos B = \dfrac{3}{5}\]
\[ \Rightarrow 3\cos A\cos B = \dfrac{3}{5}\]
\[ \Rightarrow \cos A\cos B = \dfrac{1}{5}\]
Again, using equation (1)
\[\sin A\sin B = 2\cos A\cos B\]
\[ \Rightarrow \sin A\sin B = 2*\dfrac{1}{5}\]
\[ \Rightarrow \sin A\sin B = \dfrac{2}{5}\]
Now using the formula \[\cos (A + B) = \cos A\cos B - \sin A\sin B\]
\[\cos (A + B) = \dfrac{1}{5} - \dfrac{2}{5}\]
\[ \Rightarrow \cos (A + B) = - \dfrac{1}{5}\]
Hence option C is the correct answer.
Note: The values \[\sin A\cos B\] and \[\cos A\sin B\] are not given. Thus we will give equations to find the values of \[\sin A\cos B\] and \[\cos A\sin B\].
Students must not get confused in the sum and difference formulas of cosine; they are as follows –
\[\cos (A - B) = \cos A\cos B + \sin A\sin B\]
\[\cos (A + B) = \cos A\cos B - \sin A\sin B\]
Recently Updated Pages
JEE Atomic Structure and Chemical Bonding important Concepts and Tips

JEE Amino Acids and Peptides Important Concepts and Tips for Exam Preparation

JEE Electricity and Magnetism Important Concepts and Tips for Exam Preparation

Chemical Properties of Hydrogen - Important Concepts for JEE Exam Preparation

JEE Energetics Important Concepts and Tips for Exam Preparation

JEE Isolation, Preparation and Properties of Non-metals Important Concepts and Tips for Exam Preparation

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

NEET 2025 – Every New Update You Need to Know
