
If $\tan 2\theta \tan \theta =1$, then the general value of $\theta $ is [Roorkee 1980; Karnataka CET 1992, 93, 2003]
(A) $(n+\frac{1}{2})\frac{\pi }{3}$
(B) $(n+\frac{1}{2})\pi $
(C) $(2n\pm \frac{1}{2})\frac{\pi }{3}$
(D) None of these
Answer
163.5k+ views
Hint: This is a trigonometric question. The branch of mathematics known as trigonometry studies how triangles' sides and angles correlate with one another. In order to solve the above problem, first we will take $\tan \theta $ to the opposite side and convert it into cot form, followed by $\tan ({{90}^{o}}-\theta )$ form. We will then compare the angles of both to get the value of $\theta $.
Complete step by step Solution:
The given equation is $\tan 2\theta \tan \theta =1$
Solving the equation
$\tan 2\theta =\frac{1}{\tan \theta }$
As cotangent is the inverse of tangent. So,
$\tan 2\theta =\cot \theta $
Now, we know that $\cot \theta $ can also be written as$\tan (\frac{\pi }{2}-\theta )$. Therefore,
$\tan 2\theta =\tan (\frac{\pi }{2}-\theta )$
$2\theta =n\pi +\frac{\pi }{2}-\theta $
$3\theta =n\pi +\frac{\pi }{2}$
$\theta =\frac{n\pi }{3}+\frac{\pi }{6}$
Taking $\frac{\pi }{3}$ as common from both the terms, we get the value of $\theta $ as
$\theta =(n+\frac{1}{2})\frac{\pi }{3}$
Thus, if$\tan 2\theta \tan \theta =1$, then the general value of $\theta $ is $\theta =(n+\frac{1}{2})\frac{\pi }{3}$
Therefore, the correct option is (A).
Additional Information: Trigonometric identities are equivalences in trigonometric functions. The trigonometric ratio is the relationship between the angle measurement and the side length of a right-angle triangle. Trigonometric function has widely used applications. These functions can be used for measuring the height of buildings or mountains. It can also be used in navigation.
Note: There are six different trigonometric functions. These trigonometric functions are: Sine, Cosine, Tangent, Cosecant, Secant, and Cotangent. There are various trigonometric equations showing relationships between these trigonometric functions. The trigonometric questions can be solved easily by knowing the positive or negative signs to use and doing the calculations carefully.
Complete step by step Solution:
The given equation is $\tan 2\theta \tan \theta =1$
Solving the equation
$\tan 2\theta =\frac{1}{\tan \theta }$
As cotangent is the inverse of tangent. So,
$\tan 2\theta =\cot \theta $
Now, we know that $\cot \theta $ can also be written as$\tan (\frac{\pi }{2}-\theta )$. Therefore,
$\tan 2\theta =\tan (\frac{\pi }{2}-\theta )$
$2\theta =n\pi +\frac{\pi }{2}-\theta $
$3\theta =n\pi +\frac{\pi }{2}$
$\theta =\frac{n\pi }{3}+\frac{\pi }{6}$
Taking $\frac{\pi }{3}$ as common from both the terms, we get the value of $\theta $ as
$\theta =(n+\frac{1}{2})\frac{\pi }{3}$
Thus, if$\tan 2\theta \tan \theta =1$, then the general value of $\theta $ is $\theta =(n+\frac{1}{2})\frac{\pi }{3}$
Therefore, the correct option is (A).
Additional Information: Trigonometric identities are equivalences in trigonometric functions. The trigonometric ratio is the relationship between the angle measurement and the side length of a right-angle triangle. Trigonometric function has widely used applications. These functions can be used for measuring the height of buildings or mountains. It can also be used in navigation.
Note: There are six different trigonometric functions. These trigonometric functions are: Sine, Cosine, Tangent, Cosecant, Secant, and Cotangent. There are various trigonometric equations showing relationships between these trigonometric functions. The trigonometric questions can be solved easily by knowing the positive or negative signs to use and doing the calculations carefully.
Recently Updated Pages
Geometry of Complex Numbers – Topics, Reception, Audience and Related Readings

JEE Main 2021 July 25 Shift 1 Question Paper with Answer Key

JEE Main 2021 July 22 Shift 2 Question Paper with Answer Key

JEE Main 2025 Session 2: Exam Date, Admit Card, Syllabus, & More

JEE Atomic Structure and Chemical Bonding important Concepts and Tips

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

Trending doubts
Degree of Dissociation and Its Formula With Solved Example for JEE

Instantaneous Velocity - Formula based Examples for JEE

JEE Main Chemistry Question Paper with Answer Keys and Solutions

JEE Main Reservation Criteria 2025: SC, ST, EWS, and PwD Candidates

What is Normality in Chemistry?

Chemistry Electronic Configuration of D Block Elements: JEE Main 2025

Other Pages
NCERT Solutions for Class 11 Maths Chapter 6 Permutations and Combinations

NCERT Solutions for Class 11 Maths Chapter 8 Sequences and Series

Total MBBS Seats in India 2025: Government College Seat Matrix

NEET Total Marks 2025: Important Information and Key Updates

Neet Cut Off 2025 for MBBS in Tamilnadu: AIQ & State Quota Analysis

Karnataka NEET Cut off 2025 - Category Wise Cut Off Marks
