if tan(2x)=3/4 find cos x with no computer

Question

mathmath333 · Answer

$\large\tt \color{black}{tan2x=\dfrac{3}{4}}$

$\large\tt \color{black}{\dfrac{sin2x}{cos2x}=\dfrac{3}{4}}$

$\large\tt \color{black}{\dfrac{sin^{2}2x}{cos^{2}2x}=\dfrac{9}{16}}$

$\large\tt \color{black}{\dfrac{sin^{2}2x+cos^{2}2x}{cos^{2}2x}=\dfrac{9+16}{16}}$

$\large\tt \color{black}{\dfrac{1}{cos^{2}2x}=\dfrac{25}{16}}$

$\large\tt \color{black}{cos^{2}2x=\dfrac{16}{25}}$

$\large\tt \color{black}{cos^{2}x-sin^{2}x=\dfrac{16}{25}}$

$\large\tt \color{black}{cos^{2}x=sin^{2}x+\dfrac{16}{25}}$

$\large\tt \color{black}{2cos^{2}x=cos^{2}x+sin^{2}x+\dfrac{16}{25}}$

$\large\tt \color{black}{2cos^{2}x=1+\dfrac{16}{25}}$

$\large\tt \color{black}{2cos^2x=\dfrac{41}{25}}$


$\large\tt \color{black}{cosx=\pm\dfrac{1}{5}\times\sqrt{\dfrac{41}{2}}}$