If `theta\ ` is the angle which the straight line joining the points `(x_1, y_1)a n d\ (x_2, y_2)` subtends at the origin, prove that `tantheta=(x_2y_1-x_1y_2)/(x_1x_2+y_1y_2)\ a n dcostheta=(x_1x_2+y_1y_2)/(sqrt(x1 2+y1 2x2 2+y2 2))`

Explanation

Assume` A(x_{1}, y_{1}) and B(x_{2}, y_{2})` the given points and O be the origin.


`Slope of {OA}={m}_{1}={y}_{1 times 1}`


`Slope of {OB}={m}_{2}={y}_{2 times 2}`


`tan theta=|frac{{m}_{1}-{m}_{2}}{1+{m}_{1} {~m}_{2}}|`


put the values,


`=frac{frac{{y}_{1}}{{x}_{1}}-frac{{y}_{2}}{{x}_{2}}}{1+frac{{y}_{1}}{{x}_{1}} times frac{{y}_{2}}{{x}_{2}}}`


`tan theta =frac{{x}_{2} {y}_{1}-{x}_{1} {y}_{2}}{{x}_{1} {x}_{2}+{y}_{1} {y}_{2}}`


As `cos theta=sqrt{frac{1}{1+tan ^{2} theta}}`


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