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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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