What is the general solution for 2tan^2(x)-3cot^2(x)=5 ?

The general solution for 2tan^2(x)-3cot^2(x)=5 is x=1.04719…+pin,x=2.09439…+pin

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2tan^2(x)-3cot^2(x)=5

2 tan^{2} (x) - 3 cot^{2} (x) = 5

x = 1.04719 \dots + πn, x = 2.09439 \dots + πn

Solution steps

2 tan^{2} (x) - 3 cot^{2} (x) = 5

Subtract

5

from both sides

2 tan^{2} (x) - 3 cot^{2} (x) - 5 = 0

Rewrite using trig identities

- 5 + \frac{2}{cot ^{2} ( x )} - 3 cot^{2} (x) = 0

Solve by substitution

cot (x) = 2 i, cot (x) = - 2 i, cot (x) = \frac{1}{3}, cot (x) = - \frac{1}{3}

cot (x) = 2 i :

No Solution

cot (x) = - 2 i :

No Solution

cot (x) = \frac{1}{3} : x = arccot (\frac{1}{3}) + πn

cot (x) = - \frac{1}{3} : x = arccot (- \frac{1}{3}) + πn

Combine all the solutions

x = arccot (\frac{1}{3}) + πn, x = arccot (- \frac{1}{3}) + πn

Show solutions in decimal form

x = 1.04719 \dots + πn, x = 2.09439 \dots + πn

x, 13 y = cos^{4} (1 - 2 x)

cos (x) + cos^{2} (x) + cos^{3} (x) = 0

cos (x) - sin (x) = \frac{1}{( sin ( x ) )} - \frac{1}{( cos ( x ) )}

sin^{2} (x) + cos^{5} (x) = 2

16 = 4 + 9 - 12 cos (x)

What is the general solution for 2tan^2(x)-3cot^2(x)=5 ?
The general solution for 2tan^2(x)-3cot^2(x)=5 is x=1.04719…+pin,x=2.09439…+pin