In a triangle ABC , the tangents of angles A and B are given by A = x and B = x^2+1 2x , where x > 1 . If C = 13 6 , then the value of x is:

Step-by-step solution

In any triangle ABC , A + B + C = , which implies C = - (A + B) . Taking the tangent on both sides: C = ( - (A + B)) = - (A + B) Using the tangent addition formula: C = - A + B 1 - A B = A + B A B - 1 Substitute the given values A = x and B = x^2+1 2x : A + B = x + x^2+1 2x = 2x^2 + x^2 + 1 2x = 3x^2+1 2x A B - 1 = x ( x^2+1 2x ) - 1 = x^2+1 2 - 1 = x^2-1 2 Now, substitute these into the expression for C : C = 3x^2+1 2x x^2-1 2 = 3x^2+1 x(x^2-1) = 3x^2+1 x^3-x We are given that C = 13 6 . Equating the two expressio