Questão 40 Unicamp 2021 - 1ª fase - 2º dia

Fazendo inicialmente o mmc para numerador e denominador, vem que:

$\frac{{\displaystyle \frac{\cos \theta +\mathrm{sen}\theta }{\mathrm{sen}\theta }+\frac{\cos {\displaystyle \theta }{\displaystyle -}{\displaystyle \mathrm{sen}}{\displaystyle \theta }}{\cos {\displaystyle \theta }}}}{{\displaystyle \frac{\cos \theta +\mathrm{sen}\theta }{\cos \theta }+\frac{\cos {\displaystyle \theta }{\displaystyle -}{\displaystyle \mathrm{sen}}{\displaystyle \theta }}{\mathrm{sen}{\displaystyle \theta }}}}=\frac{{\displaystyle \frac{\left(\cos {\displaystyle \theta }{\displaystyle +}{\displaystyle \mathrm{sen}}{\displaystyle \theta }\right){\displaystyle ·}{\displaystyle \cos }{\displaystyle \theta }{\displaystyle +}{\displaystyle \left(\cos \theta -\mathrm{sen}\theta \right)}{\displaystyle ·}{\displaystyle \mathrm{sen}}{\displaystyle \theta }}{\mathrm{sen}{\displaystyle \theta }{\displaystyle ·}{\displaystyle \cos }{\displaystyle \theta }}}}{{\displaystyle \frac{\left(\cos {\displaystyle \theta }{\displaystyle +}{\displaystyle \mathrm{sen}}{\displaystyle \theta }\right){\displaystyle ·}{\displaystyle \mathrm{sen}}{\displaystyle \theta }{\displaystyle +}{\displaystyle \left(\cos \theta -\mathrm{sen}\theta \right)}{\displaystyle ·}{\displaystyle \cos }{\displaystyle \theta }}{\cos {\displaystyle \theta }·\mathrm{sen}{\displaystyle \theta }}}}=$

$\frac{{\displaystyle \frac{{\cos }^2{\displaystyle \theta }{\displaystyle +}{\displaystyle \mathrm{sen}}{\displaystyle \theta }{\displaystyle ·}{\displaystyle \cos }{\displaystyle \theta }{\displaystyle +}\cos \theta ·\mathrm{sen}\theta -{\displaystyle {\mathrm{sen}}^2}{\displaystyle \theta }}{\sout{\mathrm{sen}{\displaystyle \theta }{\displaystyle ·}{\displaystyle \cos }{\displaystyle \theta }}}}}{{\displaystyle \frac{\cancel{\cos {\displaystyle \theta }·\mathrm{sen}{\displaystyle \theta }}+{\displaystyle {\mathrm{sen}}^2}{\displaystyle \theta }{\displaystyle +}{\cos }^2\theta \cancel{-\mathrm{sen}\theta ·\cos \theta }}{\sout{\mathrm{sen}{\displaystyle \theta }·\cos {\displaystyle \theta }}}}}=\frac{{\cos }^2\theta -{\mathrm{sen}}^2\theta +2·\mathrm{sen}\theta ·\cos \theta }{{\mathrm{sen}}^2\theta +{\cos }^2\theta }$

Agora, lembremos a relação fundamental da Trigonometria, e as relações de arco duplo:

• Relação fundamental da Trigonometria: ${\mathrm{sen}}^2\theta +{\cos }^2\theta =1$
• $\cos \left(2\theta \right)={\cos }^2\theta -{\mathrm{sen}}^2\theta$
• $\mathrm{sen}\left(2\theta \right)=2·\mathrm{sen}\theta ·\cos \theta$

Substituindo, a expressão dada se reduz a:

$\frac{co{s}^2\theta -se{n}^2\theta +2·sen\theta ·cos\theta }{se{n}^2\theta +co{s}^2\theta }=\frac{cos\left(2\theta \right)+sen\left(2\theta \right)}{1}=\boxed{\cos \left(2\theta \right)+\mathrm{sen}\left(2\theta \right)}$