if x = sin A cos B , y = r sin A sin B and z = r cos A , show that x^2 + y^2 + z^2 = r^2 .

if x = sin A cos B , y = r sin A sin B and z = r cos A , show that x^2

1.	if x = sin A cos B , y = r sin A sin B and z = r cos A , show that x^2 + y^2 + z^2 = r^2 .
Answer» We have,x2\xa0+ y2 + z2 = r2sin2{tex}\\alpha{/tex}cos2{tex}\\beta{/tex}\xa0+ r2sin2{tex}\\alpha{/tex}\xa0sin2{tex}\\beta{/tex}\xa0+ r2cos2{tex}\\alpha{/tex}= r2sin2{tex}\\alpha{/tex}\xa0(cos2{tex}\\beta{/tex}\xa0+ sin2{tex}\\beta{/tex}) + r2cos2{tex}\\alpha{/tex}= r2sin2{tex}\\alpha{/tex}\xa0+ r2cos2{tex}\\alpha{/tex}\xa0[{tex}\\because{/tex}\xa0cos2{tex}\\beta{/tex}\xa0+ sin2{tex}\\beta{/tex}\xa0= 1]= r2(sin2{tex}\\alpha{/tex}\xa0+ cos2{tex}\\alpha{/tex}) = r2 [{tex}\\because{/tex}\xa0sin2{tex}\\alpha{/tex}\xa0+ cos2{tex}\\alpha{/tex}\xa0= 1].Hence, (x2 + y2 + z2) = r2.