Mathematics - 3D-Coordinate Geometry Question with Solution | TestHub

Let $l,\mathrm{m}$ and $\mathrm{n}$ be the direction cosines. Then, $l=\cos \theta ,\mathrm{m}=\cos \beta ,\mathrm{n}=\cos \theta$ $$\begin{array}{l}\text{ we have }{l}^2+m^2+n^2=1\\ \Rightarrow {\cos }^2\theta +{\cos }^2\beta +{\cos }^2\theta =1\\ \Rightarrow 2{\cos }^2\theta +1-{\sin }^2\beta =1\\ \Rightarrow 2{\cos }^2\theta -{\sin }^2\beta =0\\ \Rightarrow 2{\cos }^2\theta -3{\sin }^2\beta =0\\ \left[\because {\sin }^2\beta =3{\sin }^{2}0(\text{ given })\right]\\ \Rightarrow {\tan }^2\theta =2/3\\ \therefore {\cos }^2\theta =\frac{1}{1+{\tan }^2\theta }=\frac{1}{1+2/3}=\frac{3}{5}\end{array}$$