Question:
In a right $\Delta \mathrm{ABC}$, right-angled at $\mathrm{B}$, if $\tan \mathrm{A}=1$, then verify that $2 \sin \mathrm{A} \cdot \cos \mathrm{A}=1$.
Solution:
We have,
$\tan \mathrm{A}=1$
$\Rightarrow \frac{\sin \mathrm{A}}{\cos \mathrm{A}}=1$
$\Rightarrow \sin A=\cos A$
$\Rightarrow \sin \mathrm{A}-\cos \mathrm{A}=0$
Squaring both sides, we get
$(\sin \mathrm{A}-\cos \mathrm{A})^{2}=0$
$\Rightarrow \sin ^{2} \mathrm{~A}+\cos ^{2} \mathrm{~A}-2 \sin \mathrm{A} \cdot \cos \mathrm{A}=0$
$\Rightarrow 1-2 \sin \mathrm{A} \cdot \cos \mathrm{A}=0$
$\therefore 2 \sin A \cdot \cos A=1$
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