Question

If $\sin \theta +{\sin }^2\theta =1$,
show that:
${\cos }^2\theta +{\cos }^4\theta =1$

Answer 1

Given $\sin \theta +{\sin }^2\theta =1$

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

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

squaring on both sides

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

${\cos }^4\theta +{\cos }^2\theta ={\sin }^2\theta +\sin \theta =1$ [given]

Hence Proved