2026/072) What is the value of $\sin^3 10^\circ +\sin^3 50^\circ −\sin^3 70^\circ$ equal to?

We have Formula for $\sin 3t $

$\sin 3t = 3 \sin t-4 \sin^3 t$

Putting  $10^\circ$, $10^\circ$,$70^\circ$ we get

 $\sin 30^\circ = 3\sin 10^\circ -4 \sin^3 10^\circ$

or   $\frac{1}{2} = 3 \sin 10^\circ -4 \sin^3 10^\circ\cdots(1)$

  $\sin 150^\circ = 3 \sin 50^\circ -4 \sin^3 50^\circ$

or   $\frac{1}{2} = 3 \sin 50^\circ -4 \sin^3 50^\circ\cdots(2)$

$\sin 210^\circ = 3 \sin 70^\circ -4 \sin^3 70^\circ$ 

or   $\frac{-1}{2} = 3 \sin 70^\circ -4 \sin^3 70^\circ\cdots(3)$

Adding (1) , (2) and subtracting (3) we get

$\frac{3}{2} = 3(\sin 10^\circ + \sin 50^\circ - \sin ^70^\circ) + 4(\sin^3 10^\circ +\sin^3 50^\circ −\sin^3 70^\circ) $ 

Or

$\sin^3 10^\circ +\sin^3 50^\circ −\sin^3 70^\circ = \frac{1}{4}(\frac{3}{2} -   3(\sin 10^\circ + \sin 50^\circ - \sin 70^\circ)  \cdots(1)$

Now we need to evaluate  

$\sin 10^\circ + \sin 50^\circ - \sin 70^\circ$

 Using $\sin A + \sin B = 2 \sin\frac{A+B}{2}\cos \frac{A-B}{2}$ we get

 $\sin 50^\circ + \sin 10^\circ = 2 \sin 30^\circ \cos 20^\circ$

$2 * |frac{1}{2} \cos 20^\circ$

$ \cos 20^\circ = \sin 70^\circ $

Or = $\sin 10^\circ + \sin 50^\circ - \sin 70^\circ = 0$

putting in (1) we get

 $\sin^3 10^\circ +\sin^3 50^\circ −\sin^3 70^\circ = -\frac{3}{8} $