What is the general solution for tan(x)+sec(x)=0 ?

The general solution for tan(x)+sec(x)=0 is No Solution for x\in\mathbb{R}

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tan(x)+sec(x)=0

tan (x) + sec (x) = 0

No Solution for x \in R

Solution steps

tan (x) + sec (x) = 0

Rewrite using trig identities

sec (x) + sec (x) sin (x) = 0

Factor

sec (x) + sec (x) sin (x) : sec (x) (sin (x) + 1)

sec (x) (sin (x) + 1) = 0

Solving each part separately

sec (x) = 0 or sin (x) + 1 = 0

sec (x) = 0 :

No Solution

sin (x) + 1 = 0 : x = \frac{3 π}{2} + 2 πn

Combine all the solutions

x = \frac{3 π}{2} + 2 πn

Since the equation is undefined for:

\frac{3 π}{2} + 2 πn

No Solution for x \in R

cos (θ) = \frac{3}{2}

cos^{2} (θ) + 2 sin (θ) cos (θ) - sin^{2} (θ) = 0

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

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

\frac{tan ( x )}{2} = 1

What is the general solution for tan(x)+sec(x)=0 ?
The general solution for tan(x)+sec(x)=0 is No Solution for x\in\mathbb{R}