State whether the following is 'True' or 'False' and justify your answer:
If $\cos A + \cos^2 A = 1$,then $\sin^2 A + \sin^4 A = 1$.
If $\cos A + \cos^2 A = 1$,then $\sin^2 A + \sin^4 A = 1$.
(A) True.
Given that $\cos A + \cos^2 A = 1$.
Rearranging the terms,we get $\cos A = 1 - \cos^2 A$.
Using the identity $\sin^2 A + \cos^2 A = 1$,we know that $1 - \cos^2 A = \sin^2 A$.
Therefore,$\cos A = \sin^2 A$.
Squaring both sides,we get $\cos^2 A = (\sin^2 A)^2 = \sin^4 A$.
We know that $\cos^2 A = 1 - \sin^2 A$.
Substituting this into the equation $\cos^2 A = \sin^4 A$,we get $1 - \sin^2 A = \sin^4 A$.
Rearranging the terms,we get $\sin^2 A + \sin^4 A = 1$.
Given that $\cos A + \cos^2 A = 1$.
Rearranging the terms,we get $\cos A = 1 - \cos^2 A$.
Using the identity $\sin^2 A + \cos^2 A = 1$,we know that $1 - \cos^2 A = \sin^2 A$.
Therefore,$\cos A = \sin^2 A$.
Squaring both sides,we get $\cos^2 A = (\sin^2 A)^2 = \sin^4 A$.
We know that $\cos^2 A = 1 - \sin^2 A$.
Substituting this into the equation $\cos^2 A = \sin^4 A$,we get $1 - \sin^2 A = \sin^4 A$.
Rearranging the terms,we get $\sin^2 A + \sin^4 A = 1$.
Explore More
Similar Questions
The value of the expression $\left[\operatorname{cosec}(75^{\circ}+\theta)-\sec(15^{\circ}-\theta)-\tan(55^{\circ}+\theta)+\cot(35^{\circ}-\theta)\right]$ is
Medium
View SolutionIf $\cos \theta = \frac{1}{\sqrt{2}},$ then $\theta = \ldots$ (in $^\circ$)
Easy
View SolutionIf $\sin A = \frac{1}{2},$ then the value of $\cot A$ is
Easy
View SolutionIf $3 \sin \theta = 4 \cos \theta$,then $\tan \theta = \ldots$
Easy
View SolutionIf $\sin ^{2} 35^{\circ} + \cos ^{2} \theta = 1$,then $\theta = \ldots \ldots \ldots \ldots$ (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