Question

x=xsinAcosB, y=sinAsinB z=acosA prove that x^2+y^2+z^2 = a^2

Answer 1

Answer:

Hence, X² + Y² + Z² = a²

Step-by-step explanation:

The value of X, Y and Z are:

X = a SinA CosB

Y = a SinA SinB

Z = a Cos A

The expression to be evaluated is:

$X^{2}+Y^{2}+Z^{2}$

Solve this expression as follows:

$X^{2}+Y^{2}+Z^{2}=(a\ SinA\ CosB)^{2}+(a\ SinA\ SinB)^{2}+(a\ CosA)^{2}\\=a^{2}\ Sin^{2}ACos^{2}B+a^{2}\ Sin^{2}ASin^{2}B+a^{2}\ Cos^{2}A\\=a^{2}\ Sin^{2}A[Cos^{2}B+Sin^{2}B]+a^{2}\ Cos^{2}A\\=a^{2}\ Sin^{2}A+a^{2}\ Cos^{2}A\\=a^{2}[Sin^{2}A+Cos^{2}A]\\=a^{2}$

Hence proved.