Question

Eliminate $\theta$ from the following.
$a=\sin \theta +\cos \theta ,b={\sin }^3\theta +{\cos }^3\theta$

Answer 1

$a=\sin \theta +\cos \theta .........\left(i\right)$

$b={\sin }^3\theta +{\cos }^3\theta .............\left(ii\right)$

$b⟹(\sin \theta +\cos \theta )({\sin }^2\theta -\sin \theta \cos \theta +{\cos }^2\theta )$

$=\left(a\right)(1-\sin \theta \cos \theta )$

$a^2=(\sin \theta =\cos \theta {)}^2$

$a^2=1+2\sin \theta \cos \theta$

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

Putting the value of $\sin \theta \cos \theta$ in equation $\left(ii\right)$

$b=\left(a\right)(1-\left(\frac{1-a^2}{2}\right))$

$=\left(a\right)\left[\frac{2-1+a^2}{2}\right]$

$b=\left(a\right)\left[\frac{1+a^2}{2}\right]$