What is the general solution for sec(θ)=sin^2(θ)sec(θ) ?

The general solution for sec(θ)=sin^2(θ)sec(θ) is No Solution for θ\in\mathbb{R}

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sec(θ)=sin^2(θ)sec(θ)

sec (θ) = sin^{2} (θ) sec (θ)

No Solution for θ \in R

Solution steps

sec (θ) = sin^{2} (θ) sec (θ)

Subtract

sin^{2} (θ) sec (θ)

from both sides

sec (θ) - sin^{2} (θ) sec (θ) = 0

Factor

sec (θ) - sin^{2} (θ) sec (θ) : - sec (θ) (sin (θ) + 1) (sin (θ) - 1)

- sec (θ) (sin (θ) + 1) (sin (θ) - 1) = 0

Solving each part separately

sec (θ) = 0 or sin (θ) + 1 = 0 or sin (θ) - 1 = 0

sec (θ) = 0 :

No Solution

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

sin (θ) - 1 = 0 : θ = \frac{π}{2} + 2 πn

Combine all the solutions

θ = \frac{3 π}{2} + 2 πn, θ = \frac{π}{2} + 2 πn

Since the equation is undefined for:

\frac{3 π}{2} + 2 πn, \frac{π}{2} + 2 πn

No Solution for θ \in R

sin (2 θ) + sin (θ) = 0, 0 \leq θ \leq 2 π

1 + cos (x) = 2 - cos (x)

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

2 sin^{2} (x) = sin (x) + 3

2 tan (θ) + 2 = 0

What is the general solution for sec(θ)=sin^2(θ)sec(θ) ?
The general solution for sec(θ)=sin^2(θ)sec(θ) is No Solution for θ\in\mathbb{R}