Answer to Question #188690 in Trigonometry for Killua gon

Question #188690

2 cos a cot^2 a - 6 cos a - cot^2 a = -3 is it an identity or a conditional equation? if not an identity, solve for the unknown


1
Expert's answer
2021-05-07T11:45:07-0400

The given equation 2cosa.cot2a6cosacot2a=32cosa.cot^2a-6cosa-cot^2a=-3 is identity equation.

Solving the above equation,

2cosa.cot2a6cosacot2a=32cot2a.cosacot2a+36cosa=0cot2a(2cosa1)3(2cosa1)=0(cot2a3)(2cosa1)=0(cota+3)(cota3)(2cosa1)=02cosa.cot^2a-6cosa-cot^2a=-3\newline 2cot^2a.cosa-cot^2a+3-6cosa=0\newline cot^2a(2cosa-1)-3(2cosa-1)=0\newline (cot^2a-3)(2cosa-1)=0\newline (cota+ \sqrt3)(cota-\sqrt3)(2cosa-1)=0

Then, we get

cota+3=0..............................1cota3=0..............................22cosa1=0................................3cota+ \sqrt3=0..............................1\newline cota-\sqrt3=0..............................2\newline 2cosa-1=0................................3

General solution of 1 is a=nπ+5π6a=n\pi +\frac{5 \pi }{6} , 2 is a=nπ+π6a=n\pi +\frac{\pi }{6} and 3 is a=2nπ+π3and a=2nπ+5π3a=2n\pi +\frac{\pi }{3} \text{and}\space a=2n\pi +\frac{5\pi }{3}.

Thus, the required solution are a=nπ+5π6a=n\pi +\frac{5 \pi }{6}, a=nπ+π6a=n\pi +\frac{\pi }{6}, a=2nπ+π3and a=2nπ+5π3a=2n\pi +\frac{\pi }{3} \text{and}\space a=2n\pi +\frac{5\pi }{3}.


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