What is the general solution for sec^2(x)-1+tan(x)-sqrt(3)tan(x)=sqrt(3) ?

The general solution for sec^2(x)-1+tan(x)-sqrt(3)tan(x)=sqrt(3) is x= pi/3+pin,x=(3pi)/4+pin

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sec^2(x)-1+tan(x)-sqrt(3)tan(x)=sqrt(3)

sec^{2} (x) - 1 + tan (x) - 3 tan (x) = 3

x = \frac{π}{3} + πn, x = \frac{3 π}{4} + πn

Solution steps

sec^{2} (x) - 1 + tan (x) - 3 tan (x) = 3

Subtract

3

from both sides

sec^{2} (x) - 1 + tan (x) - 3 tan (x) - 3 = 0

Rewrite using trig identities

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

Solve by substitution

tan (x) = 3, tan (x) = - 1

tan (x) = 3 : x = \frac{π}{3} + πn

tan (x) = - 1 : x = \frac{3 π}{4} + πn

Combine all the solutions

x = \frac{π}{3} + πn, x = \frac{3 π}{4} + πn

3 sec (2 x) - 2 = 0

2 sin^{2} (θ) + 3 sin (θ) = 0

sin (x + \frac{π}{6}) + sin (x - \frac{π}{6}) = - \frac{3}{2}

3 csc (\frac{2 x}{3}) - 2 = 0

cos (θ) = \frac{8}{10}

What is the general solution for sec^2(x)-1+tan(x)-sqrt(3)tan(x)=sqrt(3) ?
The general solution for sec^2(x)-1+tan(x)-sqrt(3)tan(x)=sqrt(3) is x= pi/3+pin,x=(3pi)/4+pin