If X=r sinA cosC, Y= r sinA sinC and Z= r cosA then prove that X2+Y2+Z2=r2 - Maths - Introduction to Trigonometry

Question

If X=r sinA cosC, Y= r sinA sinC and Z= r cosA then prove that
X2+Y2+Z2=r2

Pronay · Accepted Answer

$x = r sinA cosC y = r sinA sinC z = r cosA$

Consider

$x^{2} + y^{2} + z^{2} = {(r sinA cos C)}^{2} + {(r sinA sin C)}^{2} + {(r cosA)}^{2} = r^{2} \sin^{2} A \cos^{2} C+ r^{2} \sin^{2} A \sin^{2} C+ r^{2} {cos}^{2} A = r^{2} {sin}^{2} A ({cos}^{2} C+ \sin^{2} C) + r^{2} {cos}^{2} A = r^{2} \sin^{2} A (1) + r^{2} \cos^{2} A = r^{2} ({sin}^{2} A + \cos^{2} A) = r^{2}$

Therefore LHS=RHS