Question

if Cos inverse X + Cos inverse Y + Cos inverse Z is equals to pi
then prove that X square + Y square + Z square + 2 x y z is equal to 1

Answer 1

Hope it helps please mark me brainalist

Answer 2

Hence Proved.

Step-by-step explanation:

Given,

$cos^{-1} x +cos^{-1} y+cos^{-1} z=\pi$ then prove that $x^{2} +y^{2} +z^{2} +2xyz=1$

Let,

$cos^{-1}x=\alpha$ ⇒$x=cos\alpha$

$cos^{-1} y=\beta$ ⇒$y=cos \beta$

$cos^{-1} z=\gamma$ ⇒$z=cos \gamma$

∴$\alpha +\beta +\gamma=\pi$

⇒$\alpha +\beta=\pi-\gamma$

⇒$cos(\alpha +\beta)=cos(\pi-\gamma)$ by taking $cos$ on both sides.

⇒$cos\alpha.cos\beta-sin\alpha.sin\beta=-cos\gamma$  (∵$cos(\pi-\theta)=-cos\theta$)

By putting the value,

⇒$x.y-\sqrt{1-x^{2} } .\sqrt{1-y^{2} } =-z$

⇒$x.y+z=\sqrt{1-x^{2} } .\sqrt{1-y^{2} }$

⇒$x^{2} .y^{2} +z^{2} +2x.y.z=(1-x^{2} )(1-y^{2} )$

⇒$x^{2} .y^{2} +z^{2} +2x.y.z=1-y^{2} -x^{2} +x^{2} .y^{2}$

∴ $x^{2} +y^{2} +z^{2} +2x.y.z=1$

Hence Proved.