Question

In a right triangle $ABC,$right- angled at $B$, if $\tan A=1,$ then verify that $2\sin A\cos A=1$

Answer 1

In a right angled $\Delta ABC$

$\tan A=1\Rightarrow \frac{BC}{AB}=1$

$\Rightarrow BC=AB$

Let $AB=BC=k,k\in z^{+}$

By pythagoras theorem,

$A{C}^2=A{B}^2+B{C}^2\Rightarrow A{C}^2=k^2+k^2$

$\Rightarrow A{C}^2=2k^2$

$\Rightarrow AC=k\sqrt{2}$

Now $\cos A=\frac{AB}{AC}=\frac{k}{k\sqrt{2}}=\frac{1}{\sqrt{2}}$

$\sin A=\frac{BC}{AC}=\frac{k}{k\sqrt{2}}=\frac{1}{\sqrt{2}}$

$2\sin A\cos A=2\times \frac{1}{\sqrt{2}}\times \frac{1}{\sqrt{2}}=\frac{2}{2}=1$