Question

If $\frac{{\sin }^3\theta +{\cos }^3\theta }{\sin \theta +\cos \theta }+\frac{{\sin }^3\theta -{\cos }^3\theta }{\sin \theta -\cos \theta }=K$, then K =

Answer 1

We have,

$\frac{{\sin }^3\theta +{\cos }^3\theta }{\sin \theta +\cos \theta }+\frac{{\sin }^3\theta -{\cos }^3\theta }{\sin \theta -\cos \theta }=K$

$\frac{(\sin \theta +\cos \theta )({\sin }^2\theta +{\cos }^2\theta -\sin \theta \cos \theta )}{(\sin \theta +\cos \theta )}+\frac{(\sin \theta -\cos \theta )({\sin }^2\theta +{\cos }^2\theta +\sin \theta \cos \theta )}{(\sin \theta -\cos \theta )}=K$

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

$1-\sin \theta \cos \theta +1+\sin \theta \cos \theta =K$

$K=2$

Hence, this is the answer.