Solving \tan^4 x+\cot^4 x+\cot 2x=2

Lennie Davis

Lennie Davis

Answered question

2022-01-17

Solving tan4x+cot4x+cot2x=2

Answer & Explanation

lenkiklisg7

lenkiklisg7

Beginner2022-01-18Added 29 answers

Since tan4x+cot4x  and  cot2x are periodic with period π2, we will let z=tanx and assume x(0,π2) so z>0.
Then, using tan2x=2z1z2, the equation becomes
z4+1z4+1z22z=2
Multiplying out denominators and factoring, we get
(z1)(z+1)(2z6+2z4z32z22)=0
So we have one solution x=π4. The other solution corresponds to the positive, irreducible root of the sextic, and is x=0.816487

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