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

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

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Popular >

sec(x)sin^2(x)=sec(x)

Solution

sec (x) sin^{2} (x) = sec (x)

Solution

No Solution for x \in R

Solution steps

sec (x) sin^{2} (x) = sec (x)

Subtract

sec (x)

from both sides

sin^{2} (x) sec (x) - sec (x) = 0

Factor

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

sin^{2} (x) sec (x) - sec (x)

Factor out common term

sec (x)

= sec (x) (sin^{2} (x) - 1)

Factor

sin^{2} (x) - 1 : (sin (x) + 1) (sin (x) - 1)

sin^{2} (x) - 1

Rewrite

1

1^{2}

= sin^{2} (x) - 1^{2}

Apply Difference of Two Squares Formula:

x^{2} - y^{2} = (x + y) (x - y)

sin^{2} (x) - 1^{2} = (sin (x) + 1) (sin (x) - 1)

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

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

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

Solving each part separately

sec (x) = 0 or sin (x) + 1 = 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{3 π}{2} + 2 πn

sin (x) + 1 = 0

Move

1

to the right side

sin (x) + 1 = 0

Subtract

1

from 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{3 π}{2} + 2 πn

x = \frac{3 π}{2} + 2 πn

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{3 π}{2} + 2 πn, x = \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 x \in R

Graph

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Frequently Asked Questions (FAQ)

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