Question

If $\sin \theta +{\sin }^2\theta =1$, then find the value of ${\cos }^2\theta +{\cos }^4\theta$.

Answer 1

Given,

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

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

or, $\sin \theta ={\cos }^2\theta$.......(1).

Now,

${\cos }^2\theta +{\cos }^4\theta$

$={\cos }^2\theta +({\cos }^2\theta {)}^2$

$={\cos }^2\theta +(\sin \theta {)}^2$ [ Using (1)]

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

$=1$.