Calculate x, if \(\displaystyle{\tan{{\left({x}\right)}}}={\tan{{9}}}{\tan{{69}}}{\tan{{33}}}\)

Adrien Wong

Adrien Wong

Answered question

2022-03-19

Calculate x, if
tan(x)=tan9tan69tan33

Answer & Explanation

lilaznkid54rcz

lilaznkid54rcz

Beginner2022-03-20Added 2 answers

tan(x)=tan(9)tan(69)tan(33)
tan(x)=tan(39)tan(3)tan(9)
Thus,
tan2(x)=tan(3)tan(33)tan(39)tan(69)
tan2(x)=tan(3)tan(336)tan(372)tan(3+36)
tan2(x)tan(75)=tan(372)tan(336)tan(3)tan(3+36)tan(3+72)
I will not show that,
tan(5x)=tan(x72)tan(x36)tan(x)tan(x+36)tan(x+72)
Let z=cos(x)+isin(x), and w=cos(36)+isin(36) Then,
tan(x)=i(z21)2(z2+1)
Similarly we can get tan(x36),tan(x72),tan(x+72) and then multiply . We very easily get the above identity. So, we have
tan2(x)=tan2(15)x=15

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