If a-cos-cos-sin-sin-cos- b-cos-sin-sin-cos-cos-and c-sin-sin- then a2-b2-c2 is equal to

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Standard X

Mathematics

Trigonometric Identities

If a=cosαco...

Question

If $a = cos α cos β + sin α sin β cos γ$
$b = cos α sin β - sin α cos β cos γ$
and $c = sin α sin γ$ , then $a^{2} + b^{2} + c^{2}$ is equal to

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Solution

The correct option is C $1$
Given $a = cos α cos β + sin α sin β cos γ$
$b = cos α sin β - sin α cos β cos γ$
and $c = sin α sin γ$

Consider, $a^{2} + b^{2} + c^{2} = {cos}^{2} α {cos}^{2} β + {sin}^{2} α {sin}^{2} β {cos}^{2} γ + 2 cos α cos β sin α sin β cos γ + {cos}^{2} α {sin}^{2} β + {cos}^{2} β {sin}^{2} α {cos}^{2} γ - 2 cos α cos β sin α sin β c o s γ + 4 {sin}^{2} α {sin}^{2} γ$

$\Rightarrow a^{2} + b^{2} + c^{2} = {cos}^{2} α + {sin}^{2} α {cos}^{2} γ + 4 {sin}^{2} α {sin}^{2} γ$

$\Rightarrow a^{2} + b^{2} + c^{2} = 1 + 3 c^{2}$

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If a=cosαcosβ+sinαsinβcosγb=cosαsinβ−sinαcosβcosγand c=sinαsinγ, then a2+b2+c2 is equal to

If $a = cos α cos β + sin α sin β cos γ$
$b = cos α sin β - sin α cos β cos γ$
and $c = sin α sin γ$ , then $a^{2} + b^{2} + c^{2}$ is equal to