Let x2+y2+z2=r2. If tan- 1((x y/z r))+tan- 1((y z/x r))+tan- 1((z x/y r))=mπ then evaluate 2m .

Question

Let x2+y2+z2=r2. If tan- 1((x y/z r))+tan- 1((y z/x r))+tan- 1((z x/y r))=mπ then evaluate 2m . (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

tan- 1((x y/z r))+tan- 1((y z/x r))+tan- 1((z x/y r)) =tan- 1[((x y/2 r) + (y z/x r) + (z x/y r) - (x y z/r3)/1 - ((x2 + y2 + z2)r2))] x2+y2+z2=r2 ⇒ tan- 1((x y/z r))+tan- 1((y z/x r))+tan- 1((z x/y r))=(π /2) ⇒ m=(1/2)=0.5⇒ 2m=1

Q. Let x2+y2+z2=r2. If tan−1(zrxy​)+tan−1(xryz​)+tan−1(yrzx​)=mπ then evaluate 2m .

tan−1(zrxy​)+tan−1(xryz​)+tan−1(yrzx​) =tan−1[1−(r2x2+y2+z2​)2rxy​+xryz​+yrzx​−r3xyz​​] x2+y2+z2=r2 ⇒tan−1(zrxy​)+tan−1(xryz​)+tan−1(yrzx​)=2π​ ⇒m=21​=0.5⇒2m=1

Q. Let $x^{2} + y^{2} + z^{2} = r^{2} .$ If $t a n^{- 1} (\frac{x y}{zr}) + t a n^{- 1} (\frac{yz}{x r}) + t a n^{- 1} (\frac{z x}{yr}) = mπ$ then evaluate $2 m$ .