Question

$\frac{1+sin2A+cos2A}{sinA+cosA}=2cosA$
if true then write 1 and if false then write 0

Answer 1

$A={30}^{\circ }$
To Prove: $\frac{1+\sin 2A+\cos 2A}{\sin A+\cos A}=2\cos A$
L.H.S: $\frac{1+\sin 2A+\cos 2A}{\sin A+\cos A}$
= $\frac{1+\sin {60}^0+\cos {60}^0}{\sin {60}^0+\cos {60}^0}$
= $\frac{1+\frac{\sqrt{3}}{2}+\frac{1}{2}}{\frac{\sqrt{3}}{2}+\frac{1}{2}}$
= $\frac{\frac{2+\sqrt{3}+1}{2}}{\frac{\sqrt{3}+1}{2}}$
= $\frac{2+\sqrt{3}+1}{\sqrt{3}+1}$
Multiply and divide by $\sqrt{3}-1$
$\frac{(2+\sqrt{3}+1)(\sqrt{3}-1)}{(\sqrt{3}+1)(\sqrt{3}-1)}$
$\frac{2\sqrt{3}-2+3-\sqrt{3}+\sqrt{3}-1}{3-1}$
$\frac{2\sqrt{3}}{2}$
=
$\sqrt{3}$
R.H.S: $2\cos A=2\cos {30}^0=2\frac{\sqrt{3}}{2}=\sqrt{3}$