Question

If $\sin A=\frac{1}{3}$, then find the value of $\mathrm{cosAcosec}+tanAsecA$

Answer 1

$sinA=\frac{1}{3}\Rightarrow cosA=\sqrt{1-si{n}^{2}A}=\sqrt{1-\frac{1}{9}}=\sqrt{\frac{8}{9}}=\frac{2\sqrt{2}}{3}$

$tanA=\frac{sinA}{cosA}=\frac{\frac{1}{3}}{\frac{2\sqrt{2}}{3}}=\frac{1}{2\sqrt{2}}$

$secA=\frac{1}{cosA}=\frac{3}{2\sqrt{2}}$

$cosecA=\frac{1}{sinA}=3$

$cosA.cosecA+tanA.secA$

$=\frac{2\sqrt{2}}{3}.3+\frac{1}{2\sqrt{2}}.\frac{3}{2\sqrt{2}}$

$=2\sqrt{2}+\frac{3}{8}=\frac{3+16\sqrt{2}}{8}$