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If θ is the angle which the straight line joining the points (x1, y1) and (x2, y2) subtends at the origin, prove that tanθ =  \(\frac{x_2y_1 - x_1y_2}{x_1x_2 + y_1y_2}\)  and cos θ = \(\frac{x_1x_2 + y_1y_2 }{\sqrt{x_1^2 + y_1^2 }\sqrt{x_2^2 + y^2_2}}\)

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To prove:

tan θ =  \(\frac{x_2y_1 - x_1y_2}{x_1x_2 + y_1y_2}\) and cos θ = \(\frac{x_1x_2 + y_1y_2 }{\sqrt{x_1^2 + y_1^2 }\sqrt{x_2^2 + y^2_2}}\)

Assuming: 

A (x1, y1) and B (x2, y2) be the given points and O be the origin. 

Explanation:

Slope of OA = m1 = y1x

Slope of OB = m2 = y2x2 

It is given that θ is the angle between lines OA and OB.

Now, As we know that cos θ = \(\sqrt{\frac{1}{1+ tan^2θ}}\) 

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