Question

If x=r sin A cosC, y=r sin A sin C and z=r cos A, prove that r^2=x^2+y^2+z^2.

Answer 1

this is the required proof

Answer 2

$x = r \sin(a) \cos(c) \: \: \: and \: \: y = r \sin(a) \sin(c) \: and \: z = r \cos(a) \: to \: prove \: {r}^{2} = {x}^{2} + {y}^{2} + {z}^{2} \\ taking \: r.h.s. \\ {x}^{2} + {y}^{2} + {z}^{2} \\ = {(r \sin(a) \cos(c) )}^{2} + {(r \sin(a) \sin(c) ) }^{2} + {(r \cos(a)) }^{2} \\ = {r}^{2} ( { \sin(a) }^{2} { \cos(c) }^{2} + { \sin(a) }^{2} { \sin(c) }^{2} + { \cos(a) }^{2}) \\ = {r}^{2} ( { \sin(a) }^{2} ( { \cos(c) }^{2} + { \sin(c) }^{2} ) + { \cos(a) }^{2} \\ = {r}^{2} ( { \sin(a) }^{2} + { \cos(a) }^{2} ) \\ = {r}^{2} \\ proved$