Question
$
\text { If } \sin A+\cos A=\sqrt{2} \text {, prove that } \sin A \cos A=\frac{1}{2}
$
\text { If } \sin A+\cos A=\sqrt{2} \text {, prove that } \sin A \cos A=\frac{1}{2}
$
✓
Answer
We Know, $(\sin A+\cos A)^2=\sin ^2 A+\cos ^2 A+2 \sin A \cdot \cos A$ Given, $(\sin A+\cos A)=\sqrt{2}$
$
\begin{aligned}
& \Rightarrow 2=1+2 \sin \mathrm{A} \cdot \cos \mathrm{A} \\
& \Rightarrow 2 \sin \mathrm{A} \cdot \cos \mathrm{A}=1 \\
& \Rightarrow \sin \mathrm{A} \cdot \cos \mathrm{A}=\frac{1}{2}
\end{aligned}
$
$
\begin{aligned}
& \Rightarrow 2=1+2 \sin \mathrm{A} \cdot \cos \mathrm{A} \\
& \Rightarrow 2 \sin \mathrm{A} \cdot \cos \mathrm{A}=1 \\
& \Rightarrow \sin \mathrm{A} \cdot \cos \mathrm{A}=\frac{1}{2}
\end{aligned}
$
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