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

The general solution for csc^2(x)=sec(x) is x=0.90455…+2pin,x=2pi-0.90455…+2pin

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

csc^2(x)=sec(x)

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

x = 0.90455 \dots + 2 πn, x = 2 π - 0.90455 \dots + 2 πn

Solution steps

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

Subtract

sec (x)

from both sides

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

Express with sin, cos

(\frac{1}{sin ( x )})^{2} - \frac{1}{cos ( x )} = 0

Simplify

(\frac{1}{sin ( x )})^{2} - \frac{1}{cos ( x )} : \frac{cos ( x ) - sin ^{2} ( x )}{sin ^{2} ( x ) cos ( x )}

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

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

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

Add

sin^{2} (x)

to both sides

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

Square both sides

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

Subtract

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

from both sides

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

Factor

cos^{2} (x) - sin^{4} (x) : (cos (x) + sin^{2} (x)) (cos (x) - sin^{2} (x))

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

Solving each part separately

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

cos (x) + sin^{2} (x) = 0 : x = arccos (- \frac{- 1 + 5}{2}) + 2 πn, x = - arccos (- \frac{- 1 + 5}{2}) + 2 πn

cos (x) - sin^{2} (x) = 0 : x = arccos (\frac{- 1 + 5}{2}) + 2 πn, x = 2 π - arccos (\frac{- 1 + 5}{2}) + 2 πn

Combine all the solutions

x = arccos (- \frac{- 1 + 5}{2}) + 2 πn, x = - arccos (- \frac{- 1 + 5}{2}) + 2 πn, x = arccos (\frac{- 1 + 5}{2}) + 2 πn, x = 2 π - arccos (\frac{- 1 + 5}{2}) + 2 πn

Verify solutions by plugging them into the original equation

x = arccos (\frac{- 1 + 5}{2}) + 2 πn, x = 2 π - arccos (\frac{- 1 + 5}{2}) + 2 πn

Show solutions in decimal form

x = 0.90455 \dots + 2 πn, x = 2 π - 0.90455 \dots + 2 πn

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

6 cos^{3} (x) + cos^{2} (x) - 1 = 0

4 tan^{2} (x) + 12 tan (x) - 27 = 0

sin^{2} (x) - cos (x) = \frac{1}{4}

cos^{4} (a) = 3 + 4 cos^{2} (a) + cos^{4} (a)

What is the general solution for csc^2(x)=sec(x) ?
The general solution for csc^2(x)=sec(x) is x=0.90455…+2pin,x=2pi-0.90455…+2pin