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

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

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sin^2(x)*sec(x)=(sin^2(x))/(cot(x))

Solution

sin^{2} (x) \cdot sec (x) = \frac{sin ^{2} ( x )}{cot ( x )}

Solution

No Solution for x \in R

Solution steps

sin^{2} (x) sec (x) = \frac{sin ^{2} ( x )}{cot ( x )}

Subtract

\frac{sin ^{2} ( x )}{cot ( x )}

from both sides

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

Simplify

sin^{2} (x) sec (x) - \frac{sin ^{2} ( x )}{cot ( x )} : \frac{sin ^{2} ( x ) sec ( x ) cot ( x ) - sin ^{2} ( x )}{cot ( x )}

sin^{2} (x) sec (x) - \frac{sin ^{2} ( x )}{cot ( x )}

Convert element to fraction:

sin^{2} (x) sec (x) = \frac{sin ^{2} ( x ) sec ( x ) cot ( x )}{cot ( x )}

= \frac{sin ^{2} ( x ) sec ( x ) cot ( x )}{cot ( x )} - \frac{sin ^{2} ( x )}{cot ( x )}

Since the denominators are equal, combine the fractions:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{sin ^{2} ( x ) sec ( x ) cot ( x ) - sin ^{2} ( x )}{cot ( x )}

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

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

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

Factor

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

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

Factor out common term

sin^{2} (x)

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

sin^{2} (x) (sec (x) cot (x) - 1) = 0

Solving each part separately

sin^{2} (x) = 0 or sec (x) cot (x) - 1 = 0

sin^{2} (x) = 0 : x = 2 πn, x = π + 2 πn

sin^{2} (x) = 0

Apply rule

x^{n} = 0 \Rightarrow x = 0

sin (x) = 0

General solutions for

sin (x) = 0

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 = 0 + 2 πn, x = π + 2 πn

x = 0 + 2 πn, x = π + 2 πn

Solve

x = 0 + 2 πn : x = 2 πn

x = 0 + 2 πn

0 + 2 πn = 2 πn

x = 2 πn

x = 2 πn, x = π + 2 πn

sec (x) cot (x) - 1 = 0 :

No Solution

sec (x) cot (x) - 1 = 0

Express with sin, cos

- 1 + cot (x) sec (x)

Use the basic trigonometric identity:

cot (x) = \frac{cos ( x )}{sin ( x )}

= - 1 + \frac{cos ( x )}{sin ( x )} sec (x)

Use the basic trigonometric identity:

sec (x) = \frac{1}{cos ( x )}

= - 1 + \frac{cos ( x )}{sin ( x )} \cdot \frac{1}{cos ( x )}

Simplify

- 1 + \frac{cos ( x )}{sin ( x )} \cdot \frac{1}{cos ( x )} : \frac{- sin ( x ) + 1}{sin ( x )}

- 1 + \frac{cos ( x )}{sin ( x )} \cdot \frac{1}{cos ( x )}

\frac{cos ( x )}{sin ( x )} \cdot \frac{1}{cos ( x )} = \frac{1}{sin ( x )}

\frac{cos ( x )}{sin ( x )} \cdot \frac{1}{cos ( x )}

Multiply fractions:

\frac{a}{b} \cdot \frac{c}{d} = \frac{a \cdot c}{b \cdot d}

= \frac{cos ( x ) \cdot 1}{sin ( x ) cos ( x )}

Cancel the common factor:

cos (x)

= \frac{1}{sin ( x )}

= - 1 + \frac{1}{sin ( x )}

Convert element to fraction:

1 = \frac{1 sin ( x )}{sin ( x )}

= - \frac{1 \cdot sin ( x )}{sin ( x )} + \frac{1}{sin ( x )}

Since the denominators are equal, combine the fractions:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{- 1 \cdot sin ( x ) + 1}{sin ( x )}

Multiply:

1 \cdot sin (x) = sin (x)

= \frac{- sin ( x ) + 1}{sin ( x )}

= \frac{- sin ( x ) + 1}{sin ( x )}

\frac{1 - sin ( x )}{sin ( x )} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

1 - sin (x) = 0

Move

1

to the right side

1 - sin (x) = 0

Subtract

1

from both sides

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

Simplify

- sin (x) = - 1

- sin (x) = - 1

Divide both sides by

- 1

- sin (x) = - 1

Divide both sides by

- 1

\frac{- sin ( x )}{- 1} = \frac{- 1}{- 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

Since the equation is undefined for:

\frac{π}{2} + 2 πn

No Solution

Combine all the solutions

x = 2 πn, x = π + 2 πn

Since the equation is undefined for:

2 πn, π + 2 πn

No Solution for x \in R

Popular Examples

2tan(θ)+sin(θ)cos(θ)=0

2 tan (θ) + sin (θ) cos (θ) = 0

sec(x)= 25/7

sec (x) = \frac{25}{7}

sin(x/2)= 1/2 sin(x)

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

tan(θ)=(-3)/3

tan (θ) = \frac{- 3}{3}

csc(2θ)= 1/2 (sec(θ))(cos(θ))

csc (2 θ) = \frac{1}{2} (sec (θ)) (cos (θ))

Frequently Asked Questions (FAQ)

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