\begin{array} { l } { \text { If } x = r \sin A \cos C , y = r \sin A

1.	\begin{array} { l } { \text { If } x = r \sin A \cos C , y = r \sin A \sin C \text { and } z = r \cos A } \\ { \text { then prove that } x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = r ^ { 2 } } \end{array}
Answer» Given, x=rsinAcosC ..equation..1 y=rsinAsinC .....equation 2 z=rcosA .....equation 3 squaring and adding all three equations we get the following x²+y²+z²=r²(sin2Acos2C + sin2Asin2C + cos2A) =r²{sin2A(cos2C + sin2C) + cos2A} =r²{sin2A+ cos2A} ∴x²+y²+z²=r²

\begin{array} { l } { \text { If } x = r \sin A \cos C , y = r \sin A \sin C \text { and } z = r \cos A } \\ { \text { then prove that } x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = r ^ { 2 } } \end{array}

Answer»

Given, x=rsinAcosC ..equation..1

y=rsinAsinC .....equation 2

z=rcosA .....equation 3

squaring and adding all three equations we get the following

x²+y²+z²=r²(sin2Acos2C + sin2Asin2C + cos2A)

=r²{sin2A(cos2C + sin2C) + cos2A}

=r²{sin2A+ cos2A}

∴x²+y²+z²=r²