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Trigonometric Ratios Using Right Angled Triangle

If x=rsin Asi...

Question

If $x = r sin A sin B, y = r cos A sin B, z = r cos B,$ then $x^{2} + y^{2} + z^{2}$ is equal to

2 r^{2}

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r^{2}

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3 r^{2}

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r^{2} + 1

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Solution

The correct option is B $r^{2}$
$x^{2} + y^{2} + z^{2}$
$= r^{2} {sin}^{2} A {sin}^{2} B + r^{2} {cos}^{2} A {sin}^{2} B + r^{2} {cos}^{2} B$
$= r^{2} {sin}^{2} B ({sin}^{2} A + {cos}^{2} A) + r^{2} {cos}^{2} B$
$= r^{2} {sin}^{2} B + r^{2} {cos}^{2} B$
$= r^{2} ({sin}^{2} B + {cos}^{2} B)$
$= r^{2}$

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