Question

If $\sin x+{\sin }^{2}x=1$ then prove that ${\cos }^{2}x+{\cos }^{4}x=1$.

Answer 1

Given,

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

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

or,$\sin x={\cos }^{2}x$.......(1) [ ${\sin }^{2}x+{\cos }^{2}x=1$]

Now,

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

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

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

$=1$.