What is the general solution for csc^2(x)-3=6tan(x) ?

The general solution for csc^2(x)-3=6tan(x) is x=0.43008…+pin

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csc^2(x)-3=6tan(x)

csc^{2} (x) - 3 = 6 tan (x)

x = 0.43008 \dots + πn

Solution steps

csc^{2} (x) - 3 = 6 tan (x)

Square both sides

(csc^{2} (x) - 3)^{2} = (6 tan (x))^{2}

Subtract

(6 tan (x))^{2}

from both sides

(csc^{2} (x) - 3)^{2} - 36 tan^{2} (x) = 0

Rewrite using trig identities

(- 2 + cot^{2} (x))^{2} - 36 tan^{2} (x) = 0

Factor

(- 2 + cot^{2} (x))^{2} - 36 tan^{2} (x) : (- 2 + cot^{2} (x) + 6 tan (x)) (- 2 + cot^{2} (x) - 6 tan (x))

(- 2 + cot^{2} (x) + 6 tan (x)) (- 2 + cot^{2} (x) - 6 tan (x)) = 0

Solving each part separately

- 2 + cot^{2} (x) + 6 tan (x) = 0 or - 2 + cot^{2} (x) - 6 tan (x) = 0

- 2 + cot^{2} (x) + 6 tan (x) = 0 : x = arccot (- 2.17998 \dots) + πn

- 2 + cot^{2} (x) - 6 tan (x) = 0 : x = arccot (2.17998 \dots) + πn

Combine all the solutions

x = arccot (- 2.17998 \dots) + πn, x = arccot (2.17998 \dots) + πn

Verify solutions by plugging them into the original equation

x = arccot (2.17998 \dots) + πn

Show solutions in decimal form

x = 0.43008 \dots + πn

C, arctan (\frac{\frac{1}{2 \cdot π \cdot R \cdot C}}{f}) = 0

sin (x) = - 2 cos (x)

3 sin^{2} (x) - 4 cos (x) + 1 = 0

sin (x) - cos (2 x) - 1 = 0

sin^{2} (x) = 3

What is the general solution for csc^2(x)-3=6tan(x) ?
The general solution for csc^2(x)-3=6tan(x) is x=0.43008…+pin