Given a 0 and b 0, suppose arctan(a/x) + arctan(b/x) = pi/2. Show that x = sqrt(a*b). Which relation correctly gives x?

Question

Given a > 0 and b > 0, suppose arctan(a/x) + arctan(b/x) = pi/2. Show that x = sqrt(a*b). Which relation correctly gives x?

Accepted Answer

x = sqrt(a*b). Two arctangents summing to pi/2 means the angles are complementary, so their tangents are reciprocals: tan(theta1) * tan(theta2) = 1. Apply this to a/x and b/x.

Given a > 0 and b > 0, suppose arctan(a/x) + arctan(b/x) = pi/2. Show that x = sqrt(a*b). Which relation correctly gives x?

Solution

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