Question

If $\sin \theta +\cos \theta =\alpha ,$ then find the values of $|\sin \theta -\cos \theta |$ and ${\sin }^4\theta +{\cos }^4\theta .$

Answer 1

Squaring, we get
$1+\sin 2\theta =a^2\therefore \sin 2\theta =a^2-1$
$|\sin \theta -\cos \theta |=\sqrt{{(\sin \theta -\cos \theta )}^2}=\sqrt{1-\sin 2\theta }=\sqrt{2-a^2}$
${\sin }^4\theta +{\cos }^4\theta =1-2{\sin }^2\theta {\cos }^2\theta =1-\frac{1}{2}{\sin }^{2}2\theta =1-\frac{1}{2}(a^2-1{)}^2$