Question

Find the simplified value of the expression $\frac{\text{cot}A}{{\text{1+cot}}^{2}A}+\frac{\text{tan}A}{1+{\text{tan}}^{2}A}$.

Answer 1

Given equation is
$\frac{\text{cot}A}{1+{\text{cot}}^{2}A}+\frac{\text{tan}A}{1+{\text{tan}}^{2}A}$
Using the indentities,
$1+{\text{cot}}^{2}A={\text{cosec}}^{2}A$
$1+{\text{tan}}^{2}A={\text{sec}}^{2}A$
$\frac{\text{cot}A}{1+{\text{cot}}^{2}A}+\frac{\text{tan}A}{1+{\text{tan}}^{2}A}$ = $\frac{\text{Cot}A}{{\text{Cosec}}^{2}A}+\frac{\text{tan}A}{{\text{sec}}^{2}A}$

Use the identify, $\frac{1}{\text{Cosec}A}=\text{sin}A,\frac{1}{\text{secA}}=\text{cos}A$
$\frac{\text{Cot}A}{{\text{Cosec}}^{2}A}+\frac{\text{tan}A}{{\text{sec}}^{2}A}$ =
$=\text{cot}A{\text{sin}}^{2}A+\text{tan}A{\text{cos}}^{2}A$
$=\frac{\text{Cos}A{\text{sin}}^{2}A}{\text{sin}A}$ $+\frac{\text{sin}A{\text{cos}}^{2}A}{\text{cos}A}$
$=$ sinA cosA + sinA cos A
$=$ 2 sinA cosA