Question

If x = r sina cod b, y=r sin a sin b, z=r cos


a. prove that x^2 +y^2+z^2=r^2

Answer 1

See the solution in the attached file.

Answer 2

$We\:have \: x = r sin a cos a\: --(1) \\y = r sina sin b \: ---(2) \: and \\z = r cos a\: ---(3)$

$LHS =\red{ x^{2} + y^{2} + z^{2}} \\= (r sin a cos b )^{2} + ( r sin a sin b )^{2} + ( r cos a )^{2} \\= r^{2} sin^{2} a cos^{2} b + r^{2} sin^{2} a sin^{2} b + r^{2} cos^{2} a \\= r^{2} [ sin^{2} a cos^{2} b + sin^{2} a sin^{2} b + cos^{2} a] \\= r^{2} [ sin^{2} a( cos^{2} b + sin^{2} b ) + cos^{2} a ] \\= r^{2} [ sin^{2} a\times 1 + cos^{2} a ]$

$\boxed {\pink { Since, cos^{2} \theta + sin^{2} \theta = 1 }}$

$= r^{2} ( sin^{2} a+ cos^{2} a )$

$= r^{2} \times 1 \\=\green { r^{2}} \\= RHS$

$Hence\: proved$

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