Question

Prove the following statements.
$(\sin A+\cos A)(1-\sin A\cos A)={\sin }^{3}A+{\cos }^{3}A$.

Answer 1

LHS:

$(\sin A+\cos A)(1-\sin A\cos A)$

$\sin A+\cos A-{\sin }^{2}A\cos A-\sin A{\cos }^{2}A$ (1)

Substitute ${\sin }^{2}A=1-{\cos }^{2}A$

${\cos }^{2}A=1-si{n}^{2}A$

$\sin A+\cos A-{\sin }^{2}A\cos A-\sin A{\cos }^{2}A$

$\sin A+\cos A-(1-{\cos }^{2}A)\cos A-\sin A(1-{\sin }^{2}A)$

$\sin A+\cos A-(1-{\cos }^{2}A)\cos A-\sin A(1-{\sin }^{2}A)$

$\sin A+\cos A-\cos A+{\cos }^{3}A-\sin A+{\sin }^{3}A$

${\cos }^{3}A+{\sin }^{3}A$

LHS=RHS

Hence proved.