Question

In a right $∆\mathrm{ABC}$, right-angled at $\mathrm{B}$, if $\mathrm{tanA}=1$, then verify that $2\mathrm{sinA}·\mathrm{cosA}=1$.

Answer 1

$$\begin{gathered} \mathrm{We}\mathrm{have}, \\ \mathrm{tanA}=1 \\ \Rightarrow \frac{\mathrm{sinA}}{\mathrm{cosA}}=1 \\ \Rightarrow \mathrm{sinA}=\mathrm{cosA} \\ \Rightarrow \mathrm{sinA}-\mathrm{cosA}=0 \\ \mathrm{Squaring}\mathrm{both}\mathrm{sides},\mathrm{we}\mathrm{get} \\ {\left(\mathrm{sinA}-\mathrm{cosA}\right)}^2=0 \\ \Rightarrow {\sin }^2\mathrm{A}+{\cos }^2\mathrm{A}-2\mathrm{sinA}·\mathrm{cosA}=0 \\ \Rightarrow 1-2\mathrm{sinA}·\mathrm{cosA}=0 \\ \therefore 2\mathrm{sinA}·\mathrm{cosA}=1 \end{gathered}$$