What is the general solution for 1/(6tan^6(x))= 1/(6sec^6(x)) ?

The general solution for 1/(6tan^6(x))= 1/(6sec^6(x)) is No Solution for x\in\mathbb{R}

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1/(6tan^6(x))= 1/(6sec^6(x))

\frac{1}{6 tan ^{6} ( x )} = \frac{1}{6 sec ^{6} ( x )}

No Solution for x \in R

Solution steps

\frac{1}{6 tan ^{6} ( x )} = \frac{1}{6 sec ^{6} ( x )}

Subtract

\frac{1}{6 sec ^{6} ( x )}

from both sides

\frac{1}{6 tan ^{6} ( x )} - \frac{1}{6 sec ^{6} ( x )} = 0

Simplify

\frac{1}{6 tan ^{6} ( x )} - \frac{1}{6 sec ^{6} ( x )} : \frac{sec ^{6} ( x ) - tan ^{6} ( x )}{6 tan ^{6} ( x ) sec ^{6} ( x )}

\frac{sec ^{6} ( x ) - tan ^{6} ( x )}{6 tan ^{6} ( x ) sec ^{6} ( x )} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

sec^{6} (x) - tan^{6} (x) = 0

Factor

sec^{6} (x) - tan^{6} (x) : (sec (x) + tan (x)) (sec^{2} (x) - sec (x) tan (x) + tan^{2} (x)) (sec (x) - tan (x)) (sec^{2} (x) + sec (x) tan (x) + tan^{2} (x))

(sec (x) + tan (x)) (sec^{2} (x) - sec (x) tan (x) + tan^{2} (x)) (sec (x) - tan (x)) (sec^{2} (x) + sec (x) tan (x) + tan^{2} (x)) = 0

Rewrite using trig identities

(sec^{2} (x) + tan^{2} (x) + sec (x) tan (x)) (sec^{2} (x) + tan^{2} (x) - sec (x) tan (x)) = 0

Solving each part separately

sec^{2} (x) + tan^{2} (x) + sec (x) tan (x) = 0 or sec^{2} (x) + tan^{2} (x) - sec (x) tan (x) = 0

sec^{2} (x) + tan^{2} (x) + sec (x) tan (x) = 0 :

No Solution

sec^{2} (x) + tan^{2} (x) - sec (x) tan (x) = 0 :

No Solution

Combine all the solutions

No Solution for x \in R

cos^{2} (2 x) - 2 sin^{2} (x) - 1 = 0

cos^{2} (x) + 2 = sin (x)

- sin (2 x) - 3 cos (x) = 0

x, y = 3 cos (f xx + \frac{π}{2}) + 5

sin (x) cos (x) = sin (x), 0 < x \leq 2 π

What is the general solution for 1/(6tan^6(x))= 1/(6sec^6(x)) ?
The general solution for 1/(6tan^6(x))= 1/(6sec^6(x)) is No Solution for x\in\mathbb{R}