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Mathematics Question on Some Applications of Trigonometry

Let x,yx, y and zz be positive real numbers Suppose x,yx, y and zz are the lengths of the sides of a triangle opposite to its angles X,YX, Y and ZZ, respectively If tan⁡x2+tan⁡z2=2yx+y+z\tan \frac{x}{2}+\tan \frac{z}{2}=\frac{2 y}{x+y+z} then which of the following statements is/are TRUE?

A

2Y=X+Z2 Y=X+Z

B

Y=X+ZY=X+Z

C

tan⁡X2=xy+z\tan \frac{X}{2}=\frac{x}{y+z}

D

x2+z2−y2=xzx^{2}+z^{2}-y^{2}=x z

Answer

Y=X+ZY=X+Z

Explanation

Solution

(B) Y=X+ZY=X+Z
(C) tan⁡X2=xy+z\tan \frac{X}{2}=\frac{x}{y+z}