Question

If x = r cos A sin B, y = r cos A cosB and z = r sinA, then prove that x^2+y^2+z^2=r^2​

Answer 1

Note: sin²x + cos²x = 1

Step-by-step explanation:

Square on both sides of x, y and z:

x² = (rcosAsinB)² = r²cos²Asin²B

y² = (rcosAcosB)² = r²cos²Acos²B

z² = (rsinA)² = r²sin²A

Add x² and y²:

=> x² + y² = r²cos²Asin²B + r²cos²Acos²B

= r²cos²A(sin²B + cos²B)

= r²cos²A(1)

= r²cos²A

Now, adding (x² + y²) and z²:

=> x² + y² + z² = r²cos²A + r²sin²A

=> x² + y² + z² = r²(cos²A + sin²A) = r²(1)

=> x² + y² + z² = r², as desired