Question

Fatore a expressão

(transforme em produto de razões trigonométricas)

tg(π/5) − tg(π/7) − tg(2π/35)

Answer 1

Olá Lukyo.


Identidade trigonométrica utilizada:

$\star~~\boxed{\boxed{\mathsf{tg(\alpha\pm\beta)=\dfrac{tg(\alpha)\pm tg(\beta)}{1\mp tg(\alpha)\cdot tg(\beta)}}}}$

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Organiando e desenvolvendo a expressão

$\mathsf{tg\Big(\dfrac{\pi}{5}\Big)-tg\Big(\dfrac{\pi}{7}\Big)-tg\Big(\dfrac{2\pi}{35}\Big)}$

Da identidade temos que:

$\mathsf{tg\Big(\dfrac{\pi}{5}-\dfrac{\pi}{7}\Big)=\dfrac{tg\Big(\dfrac{\pi}{5}\Big)-tg\Big(\dfrac{\pi}{7}\Big)}{1+tg\Big(\dfrac{\pi}{5}\Big)\cdot tg\Big(\dfrac{\pi}{7}\Big)}}}\\\\=\\\\\mathsf{tg\Big(\dfrac{2\pi}{35}\Big) + tg\Big(\dfrac{2\pi}{35}\Big)\cdot tg\Big(\dfrac{\pi}{5}\Big)\cdot tg\Big(\dfrac{\pi}{7}\Big)=tg\Big(\dfrac{\pi}{5}\Big)-tg\Big(\dfrac{\pi}{7}\Big)}$

Substituindo:

$\mathsf{tg\Big(\dfrac{2\pi}{35}\Big) + tg\Big(\dfrac{2\pi}{35}\Big)\cdot tg\Big(\dfrac{\pi}{5}\Big)\cdot tg\Big(\dfrac{\pi}{7}\Big) - tg\Big(\dfrac{2\pi}{35}\Big)}\\\\=\\\\\mathsf{tg\Big(\dfrac{2\pi}{35}\Big)\cdot tg\Big(\dfrac{\pi}{5}\Big)\cdot tg\Big(\dfrac{\pi}{7}\Big)}$


Dúvidas? comente.