Question

Solve for x, if:

tan (cos^-1x) = ${\displaystyle \frac{2}{\sqrt{5}}}$

Answer 1

tan (cos^-1x) = ${\displaystyle \frac{2}{\sqrt{5}}}$

⇒  cos^-1 x = tan^-1${\displaystyle \left(\frac{2}{\sqrt{5}}\right)}$

⇒  tan^-1${\displaystyle \left(\frac{\sqrt{1-{\text{x}}^2}}{\text{x}}\right)}$ = tan^-1${\displaystyle \left(\frac{2}{\sqrt{5}}\right)}$ 

⇒  ${\displaystyle \frac{\sqrt{1-{\text{x}}^2}}{\text{x}}=\frac{2}{\sqrt{5}} \Rightarrow \frac{1-{\text{x}}^2}{{\text{x}}^2}=\frac{4}{5}}$

⇒  5 - 5x² = 4x²                  ⇒     5 = 9x² 

^⇒  ${\displaystyle \frac{5}{9}=x^2 \Rightarrow x=\pm \frac{\sqrt{5}}{3}}$