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Giovana Martins escreveu:

[latex]\mathrm{Identidade\ trigonom\acute{e}trica:sin\left ( \frac{\pi }{2}-x \right ) = cos(x)}[/latex]

[latex]\mathrm{Pois\ sin(\alpha -\beta ) = sin(\alpha ) cos(\beta ) - sin(\beta )cos(\alpha )\ \therefore\ (\alpha ,\beta ) = \left ( \frac{\pi }{2},x \right ) \therefore\ sin\left ( \frac{\pi }{2}-x \right ) = cos(x)}[/latex]

[latex]\mathrm{Item\ A ) \ sin( 80^{\circ})+cos( 50^{\circ} )= sin( 80^{\circ}) + sin ( 40^{\circ} ) = 2 sin\left ( \frac{80^{\circ} + 40^{\circ}}{2} \right ) cos\left ( \frac{80^{\circ} - 40^{\circ}}{2} \right ) = \sqrt{3}cos ( 20^{\circ} ) }[/latex]

[latex]\mathrm{No\ item\ A ) \ foi\ utilizado\ Prostaf\acute{e}rese\ sendo\ sin ( \alpha ) + sin ( \beta ) = 2sin\left ( \frac{\alpha +\beta }{2} \right ) cos\left ( \frac{\alpha -\beta }{2} \right ) }[/latex]

[latex]\mathrm{Item\ B ) \ tan ( 25^{\circ} ) + tan ( 65^{\circ} ) = \frac{sin(25^{\circ})}{cos ( 25^{\circ} ) } 

 + \frac{sin ( 65^{\circ} )}{cos ( 65^{\circ} ) }=\frac{sin ( 25^{\circ} ) cos ( 65^{\circ} ) + sin ( 65^{\circ} ) cos ( 25^{\circ} ) }{cos ( 25^{\circ} ) cos ( 65^{\circ} ) }}[/latex]

[latex]\mathrm{Assim : tan ( 25^{\circ} ) + tan ( 65^{\circ} ) = \frac{sin ( 25^{\circ} + 65^{\circ} ) }{ cos ( 25^{\circ} ) cos ( 65^{\circ} ) } = \frac{1}{cos ( 25^{\circ} ) cos ( 65^{\circ} ) }}[/latex]

[latex]\mathrm{Por\ Prostaf\acute{e}rese:cos \left ( \frac{\alpha +\beta }{2} \right ) cos\left ( \frac{\alpha -\beta }{2} \right ) = \frac{cos (\alpha ) + cos ( \beta ) }{2}}[/latex]

[latex]\mathrm{Logo\ \frac{\alpha +\beta }{2} = 65^{\circ}\ (i)\ e\ \frac{\alpha -\beta }{2} = 25^{\circ}\ (ii)\ \therefore\ De\ (i)\ e\ (ii) : (\alpha ,\beta ) = ( 90^{\circ},40^{\circ} ) }[/latex]

[latex]\mathrm{\therefore\ tan(25^{\circ} ) + tan(65^{\circ} ) = \frac{1}{cos ( 25^{\circ} ) cos(65^{\circ} ) } = \frac{2}{cos ( 90^{\circ} ) + cos ( 40^{\circ} ) } = 2sec ( 40^{\circ} ) }[/latex]

[latex]\mathrm{Item\ C)\ 1-sin (x) = sin\left ( \frac{\pi}{2} \right )-sin  ( x ) = 2sin\left ( \frac{\pi }{4}-\frac{x}{2} \right ) cos\left ( \frac{\pi }{4}+\frac{x}{2} \right )}[/latex]