What is the general solution for sec(x)sin(x)+sec(x)=2sec(x) ?

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

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sec(x)sin(x)+sec(x)=2sec(x)

sec (x) sin (x) + sec (x) = 2 sec (x)

No Solution for x \in R

Solution steps

sec (x) sin (x) + sec (x) = 2 sec (x)

Subtract

2 sec (x)

from both sides

sec (x) sin (x) - sec (x) = 0

Factor

sec (x) sin (x) - sec (x) : sec (x) (sin (x) - 1)

sec (x) sin (x) - sec (x)

Factor out common term

sec (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

sec (x) = 0

sec (x) \leq - 1 or sec (x) \geq 1

No Solution

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

sin (x) - 1 = 0

Move

1

to the right side

sin (x) - 1 = 0

Add

1

to both sides

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

Simplify

sin (x) = 1

sin (x) = 1

General solutions for

sin (x) = 1

sin (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

x = \frac{π}{2} + 2 πn

x = \frac{π}{2} + 2 πn

Combine all the solutions

x = \frac{π}{2} + 2 πn

Since the equation is undefined for:

\frac{π}{2} + 2 πn

No Solution for x \in R

3 - sin (x) (4 + 6 sin (x)) = 0

cos (a) = - \frac{3}{5}, sin (a) > 0

cot (x) - cot (33.4 7^{\circ}) = \frac{1485}{2460}

3 + \frac{1}{5} cos (\frac{θ}{3} + \frac{7 π}{4}) = 3

sin (θ) = - \frac{13}{7}

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