Question

Solve : ${\cos }^{6}A-{\sin }^{6}A=\cos 2A(1-\frac{1}{4}{\sin }^{2}2A)$

Answer 1

${\cos }^{6}A-{\sin }^{6}A$

$=({\cos }^{2}A{)}^3-({\sin }^{2}A{)}^3$

$=[{\cos }^{2}A-{\sin }^{2}A][{\cos }^{4}A+{\cos }^{2}A{\sin }^{2}A+{\sin }^{4}A]$

$=\cos 2A[{\cos }^{4}A+{\sin }^{4}A+2{\cos }^{2}A{\sin }^{2}A-{\cos }^{2}A{\sin }^{2}A]$

$=\cos 2A\left[\right({\cos }^{2}A-{\sin }^{2}A{)}^2-{\sin }^{2}A{\cos }^{2}A]$

$=\cos 2A(1-\frac{4{\sin }^{2}A{\cos }^{2}A}{4})$

$=\cos 2A(1-\frac{(2{\sin }^{2}A{\cos }^{2}A{)}^2}{4})$

$=\cos 2A(1-\frac{{\sin }^{2}2A}{4})$