Question

If $\sin \theta -\cos \theta =1$, then the value of ${\sin }^3\theta -{\cos }^3\theta$ is ________.

Answer 1

We have,

$\sin \theta -\cos \theta =1$ ……. (1)

On squaring both sides, we get

${\sin }^2\theta +{\cos }^2\theta -2\sin \theta \cos \theta =1$

$1-2\sin \theta \cos \theta =1$

$\sin \theta \cos \theta =0$

Since,

$\Rightarrow {\sin }^3\theta -{\cos }^3\theta$

$\Rightarrow (\sin \theta -\cos \theta )({\sin }^2\theta +{\cos }^2\theta +\sin \theta \cos \theta )[\because x^3-y^3=(x-y)(x^2+y^2+xy)]$

$\Rightarrow \left(1\right)(1+0)$

$\Rightarrow 1$

Hence, this is the answer.