Let x ,   y and z be positive real numbers. Suppose x ,   y and z are the lengths of the sides of a triangle opposite to its angles X ,   Y and Z respectively. If tan X 2 + tan Z 2 = 2 y x + y + z , then which of the following statements is/are TRUE?
C. tan X 2 = x y + z
We have, tan X 2 + tan Z 2 = 2 y x + y + z   &   s = x + y + z = perimeter of a triangle Apply half-angle formula and x + y + z = s then, ⇒ ∆ s s − x + ∆ s s − z = y s ⇒ ∆ = s − x s − z ⇒ ∆ 2 = s s − x s − y s − z = s − x 2 s − z 2 ⇒ y 2 = x 2 + z 2 ⇒ ∠ Y = 90 ° ⇒ ∠ Y = ∠ X + ∠ Z An option B is correct. Now, x y + z = sin X sin Y + sin Z = sin X 1 + sin π 2 - X = 2 sin X 2 cos X 2 2 cos 2 X 2 = tan X 2 An Option C is correct.
Related: Mathematics — Properties of Triangles · All PYQ Banks