Question

If x = r sinA cosB , y = r sinA sinB and z = r cosA , prove that   ${\displaystyle x^2+y^2+z^2=r^2}$

Answer 1

LHS = ${\displaystyle {\left(r\sin A\cos B\right)}^2+{\left(r\sin A\sin B\right)}^2+{\left(r\cos A\right)}^2}$

⇒ ${\displaystyle r^2{\sin }^{2}A{\cos }^{2}B+r^2{\sin }^{2}A{\sin }^{2}B+r^2{\cos }^{2}A}$

⇒ ${\displaystyle r^2{\sin }^{2}A({\cos }^{2}B+{\sin }^{2}B)+r^2{\cos }^{2}A}$

⇒ ${\displaystyle r^2({\sin }^{2}A+{\cos }^{2}A)=r^2}$ = RHS