Question

If $x=rsinAcosB,y=rsinAsinBandz=rcosA,thenprovethat:x^2+y^2+z^2=r^2$

Answer 1

x = r sin A cos B =>$x^2=r^{2}si{n}^{2}Aco{s}^{2}B$

y = r sin A sin B =>$y^2=r^{2}si{n}^{2}Asi{n}^{2}B$

z = r cos A => $z^2=r^{2}co{s}^{2}A$

=>$x^2+y^2+z^2=r^{2}si{n}^{2}Aco{s}^{2}B+r^{2}si{n}^{2}Asi{n}^{2}B+r^{2}co{s}^{2}A=r^2\left(si{n}^{2}A\right(co{s}^{2}B+si{n}^{2}B)+co{s}^{2}A)=r^2(si{n}^{2}A+co{s}^{2}A)=r^2$