Question

If x = a cosα cosβ , y = a cosα sinβ and z = a sinα , show that x² + y² + z² = a²
(α = alpha β = beta, just for information)

Answer 1

Answer:

there is the answer glad to help

Answer 2

Given :

An equation $x=acos\alpha cos\beta ,y=cos\alpha sin\beta ,z=asin\alpha$

To prove :

$x^{2} +y^{2} +z^{2} =a^{2}$

Step-by-step explanation:

• These steps can be proved by following these steps.
• Firstly , $x^{2}$ must be calculated.

       $x^{2} =a^{2} cos^{2} \alpha cos^{2} \beta$

• Secondly, $y^{2}$ must be calculated.

       $y^{2} =a^{2} cos^{2} \alpha sin^{2} \beta$

• Thirdly, $z^{2}$ must be calculated.

       $z^{2} =a^{2} sin^{2} \alpha$

• Sum all the terms $x^{2} +y^{2} +z^{2}$.

       $x^{2} +y^{2} +z^{2} =a^{2} cos^{2} \alpha cos^{2} \beta+a^{2} cos^{2} \alpha sin^{2} \beta+a^{2} sin^{2} \alpha$

• Take out $a^{2}$ as common.

       $x^{2} +y^{2} +z^{2} =a^{2}[ cos^{2} \alpha cos^{2} \beta+ cos^{2} \alpha sin^{2} \beta+ sin^{2} \alpha]$

• Take $cos^{2} \alpha$ as common from 1st and 2nd terms

       $x^{2} +y^{2} +z^{2} =a^{2}[ cos^{2} \alpha (cos^{2} \beta+ sin^{2} \beta)+ sin^{2} \alpha]$

• Formula that can be used is,

      $sin^{2} x+cos^{2} x=1$

• By using this formula we get,

      $x^{2} +y^{2} +z^{2} =a^{2}[ cos^{2} \alpha (1)+ sin^{2} \alpha]$

• By applying this formula again,

       $x^{2} +y^{2} +z^{2} =a^{2}[1]$

Final answer :

$x^{2} +y^{2} +z^{2} =a^{2}$ where $x=acos\alpha cos\beta ,y=cos\alpha sin\beta ,z=asin\alpha$