Question

$A={\cos }^2\theta +{\sin }^4\theta$, find range of $A$

Answer 1

$A={\cos }^2\theta +{\sin }^4\theta ..........\left(I\right)$

using ${\cos }^2\theta +{\sin }^2\theta =\mathrm{1...............}\left(II\right)$

${\cos }^2\theta =1-{\sin }^2\theta ..............\left(III\right)$

$A=1-{\sin }^2\theta +{\sin }^4\theta$ using equation $\left(II\right)$

$-1-{\sin }^2\theta (1-{\sin }^2\theta )$ using equation $\left(III\right)$

$=1-{\sin }^2\theta {\cos }^2\theta$

$=1-\frac{4{\sin }^2\theta {\cos }^2\theta }{4}\{\because \sin 2A=2\sin A\cos A\}$

$A=1-\frac{{\sin }^2\theta {\cos }^2\theta }{4}$

when $\theta =0A=q\sin \theta =0$

when $\theta =\frac{\pi }{4}A=1-\frac{1}{4}\sin \frac{\pi }{2}=1$

$=3/4$

when $\theta =\pi /2A=1\sin \pi =0$

from above $\frac{3}{4}\le A\le 1$