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[tex] \cos( \alpha ) = x[/tex]
[tex] {sin}^{2}( \alpha ) = 1 - {cos}^{2}(x) [/tex]
[tex] {sin}^{2}( \alpha ) = 1 - ({x})^{2} [/tex]
[tex] \sin( \alpha ) = ± \sqrt{1 - {x}^{2} } [/tex]
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Look :
[tex] \sin(2 \alpha ) = 2. \sin( \alpha ) . \cos( \alpha ) [/tex]
[tex] \sin(2 \alpha ) = 2 \times ( ± \sqrt{1 - {x}^{2} })(x) \\ [/tex]
[tex] \sin(2 \alpha ) = ± \: 2 \: x \: \sqrt{1 - {x}^{2} } [/tex]
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[tex] \cos(2 \alpha ) = {cos}^{2}(x) - {sin}^{2}(x) [/tex]
[tex] \cos(2 \alpha ) = {x}^{2} - ({± \: \sqrt{1 - {x}^{2} } })^{2} \\ [/tex]
[tex] \cos(2 \alpha ) = {x}^{2} - |1 - {x}^{2} | [/tex]
If | 1 - x² | ≥ 0 :
[tex] \cos(2 \alpha ) = {x}^{2} - (1 - {x}^{2}) [/tex]
[tex] \cos(2 \alpha ) = {x}^{2} - 1 + {x}^{2} [/tex]
[tex] \cos(2 \alpha ) = 2 {x}^{2} - 1 [/tex]
If | 1 - x² | < 0 :
[tex] \cos(2 \alpha ) = {x}^{2} - ( - 1 )(1 - {x}^{2} ) \\ [/tex]
[tex] \cos(2 \alpha ) = {x}^{2} + 1 - {x}^{2} [/tex]
[tex] \cos(2 \alpha ) = 1 [/tex]
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[tex] {sin}^{2}( \frac{ \alpha }{2} ) = \frac{1 - \cos( \alpha ) }{2} \\ [/tex]
[tex] {sin}^{2}( \frac{ \alpha }{2} ) = \frac{1 - x}{2} \\ [/tex]
[tex] \sin( \frac{ \alpha }{2} ) = ± \sqrt{ \frac{1 - x}{2} } \\ [/tex]
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[tex] {cos}^{2}( \frac{ \alpha }{2} ) = \frac{1 + \cos( \alpha ) }{2} \\ [/tex]
[tex] {cos}^{2}( \frac{ \alpha }{2} ) = \frac{1 + x}{2} \\ [/tex]
[tex] \cos( \frac{ \alpha }{2} ) = ± \sqrt{ \frac{1 + x}{2} } \\ [/tex]
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Thus the correct answer is (( D )) .
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