Write 'True' or 'False' and justify your answer.
$(\tan \theta+2)(2 \tan \theta+1)=5 \tan \theta+\sec ^{2} \theta$
$(\tan \theta+2)(2 \tan \theta+1)=5 \tan \theta+\sec ^{2} \theta$
(B) False.
$L$.$H$.$S$. $= (\tan \theta + 2)(2 \tan \theta + 1)$
$= 2 \tan^2 \theta + \tan \theta + 4 \tan \theta + 2$
$= 2 \tan^2 \theta + 5 \tan \theta + 2$
Using the identity $\tan^2 \theta = \sec^2 \theta - 1$:
$= 2(\sec^2 \theta - 1) + 5 \tan \theta + 2$
$= 2 \sec^2 \theta - 2 + 5 \tan \theta + 2$
$= 2 \sec^2 \theta + 5 \tan \theta$
Since $2 \sec^2 \theta + 5 \tan \theta \neq 5 \tan \theta + \sec^2 \theta$,the given statement is False.
$L$.$H$.$S$. $= (\tan \theta + 2)(2 \tan \theta + 1)$
$= 2 \tan^2 \theta + \tan \theta + 4 \tan \theta + 2$
$= 2 \tan^2 \theta + 5 \tan \theta + 2$
Using the identity $\tan^2 \theta = \sec^2 \theta - 1$:
$= 2(\sec^2 \theta - 1) + 5 \tan \theta + 2$
$= 2 \sec^2 \theta - 2 + 5 \tan \theta + 2$
$= 2 \sec^2 \theta + 5 \tan \theta$
Since $2 \sec^2 \theta + 5 \tan \theta \neq 5 \tan \theta + \sec^2 \theta$,the given statement is False.
Explore More
Similar Questions
If $A, B$ and $C$ are interior angles of a triangle $ABC$,then show that $\sin \left(\frac{B+C}{2}\right) = \cos \frac{A}{2}$.
Medium
View SolutionProve the following identity,where the angles involved are acute angles for which the expressions are defined:
$\frac{\cos A}{1+\sin A}+\frac{1+\sin A}{\cos A}=2 \sec A$
$\frac{\cos A}{1+\sin A}+\frac{1+\sin A}{\cos A}=2 \sec A$
Medium
View SolutionProve that $\sqrt{\sec ^{2} \theta+\operatorname{cosec}^{2} \theta}=\tan \theta+\cot \theta$
Difficult
View SolutionProve the following identity,where the angles involved are acute angles for which the expressions are defined:
$(\operatorname{cosec} \theta - \cot \theta)^2 = \frac{1 - \cos \theta}{1 + \cos \theta}$
$(\operatorname{cosec} \theta - \cot \theta)^2 = \frac{1 - \cos \theta}{1 + \cos \theta}$
Medium
View SolutionIf $\tan ^{2} \theta = \sin ^{2} \theta + \cos ^{2} \theta$,then $\theta = \ldots$ (in $^{\circ}$)
Easy
View SolutionVedclass Products
For Students
Vedclass Test Series
Mock tests in real JEE/NEET style with performance analysis. 5-day free trial.
Start Free TrialFor Teachers
Exam Paper Generator
Generate Set A/B/C/D exam papers from 7.5L+ questions in 2 minutes. 3 chapters free.
Try FreeFor Institutes
Online Exam Module
Live online exams with unlimited students, 360° analytics & white-label branding.
See Demo