Question

If $t_n={\sin }^n\theta +{\cos }^n\theta$, then prove that $\frac{t_3-t_5}{t_1}=\frac{t_5-t_7}{t_3}$

Answer 1

$L.H.S.$ $[({\sin }^3\theta +{\cos }^3\theta )-({\sin }^5\theta +{\cos }^5\theta )]+(\sin \theta +\cos \theta )$
$=[{\sin }^3\theta (1-{\sin }^2\theta )+{\cos }^3\theta (1-{\cos }^2\theta )]+(\sin \theta +\cos \theta )$
$={\sin }^2\theta {\cos }^2\theta [\sin \theta +\cos \theta ]+[\sin \theta +\cos \theta ]$
$={\sin }^2\theta {\cos }^2\theta$.
Similarly, $R.H.S.$ $={\sin }^2\theta {\cos }^2\theta$.