Question

If cos A=2x/1+xsquare then find the values of sinA and tanA in terms of x

Answer 1

Given:

$cosA=\frac{2x}{1+x^2}$

$\text{Take, x=tan\theta }$

$cosA=\frac{2\,tan\theta}{1+tan^2\theta}$

$cosA=sin2\theta$

$cosA=cos(90^{\circ}-2\theta)$

$\implies\;A=90^{\circ}-2\theta$

Now,

$sinA$

$=sin(90^{\circ}-2\theta)$

$=cos2\theta$

$=\frac{1-tan^2\theta}{1+tan^2\theta}$

$=\frac{1-x^2}{1+x^2}$

$\implies\:\boxed{\bf\,sinA=\frac{1-x^2}{1+x^2}}$

$tanA$

$=tan(90^{\circ}-2\theta)$

$=cot2\theta$

$=\frac{1}{tan2\theta}$

$=\frac{1}{2tan\theta/1-tan^2\theta}$

$=\frac{1-tan^2\theta}{2tan\theta}$

$=\frac{1-x^2}{2x}$

$\implies\:\boxed{\bf\,tanA=\frac{1-x^2}{2x}}$