Question

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

Answer 1

$sin\theta +si{n}^2\theta =1\to given$

$co{s}^2\theta +co{s}_4\theta =1\to Toprove$

LHS:

$co{s}^2\theta +co{s}^4\theta$

$co{s}^2\theta +\left(co{s}^2\theta \right)$

$co{s}^2\theta +(1-si{n}^2)\theta$

$co{s}^2\theta +1-2si{n}^2\theta +si{n}^4\theta$

$1-si{n}^2\theta +1-2si{n}^2\theta +si{n}^4\theta$

$2-3si{n}^2\theta +si{n}^4\theta$

$2-3si{n}^2\theta +(si{n}^2\theta {)}^2$

$2-3si{n}^2\theta +(si{n}^2\theta {)}^2$

$2-3(1-sin\theta )+(1-sin\theta {)}^2$

$2-3+3sin\theta +1-2sin\theta +si{n}^2\theta$

$2-2+sin\theta +si{n}^2\theta$

0+1

=1 RHS.