What is the general solution for (sec(x))/(cos(x))-1=0 ?

The general solution for (sec(x))/(cos(x))-1=0 is x=2pin,x=pi+2pin

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(sec(x))/(cos(x))-1=0

Solution

\frac{sec ( x )}{cos ( x )} - 1 = 0

Solution

x = 2 πn, x = π + 2 πn

Degrees

x = 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n

Solution steps

\frac{sec ( x )}{cos ( x )} - 1 = 0

Simplify

\frac{sec ( x )}{cos ( x )} - 1 : \frac{sec ( x ) - cos ( x )}{cos ( x )}

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

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

Multiply:

1 \cdot cos (x) = cos (x)

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

\frac{sec ( x ) - cos ( x )}{cos ( x )} = 0

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

sec (x) - cos (x) = 0

Rewrite using trig identities

- cos (x) + sec (x)

Use the basic trigonometric identity:

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

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

- \frac{1}{sec ( x )} + sec (x) = 0

Solve by substitution

- \frac{1}{sec ( x )} + sec (x) = 0

Let:

sec (x) = u

- \frac{1}{u} + u = 0

- \frac{1}{u} + u = 0 : u = 1, u = - 1

- \frac{1}{u} + u = 0

Multiply both sides by

u

- \frac{1}{u} + u = 0

Multiply both sides by

u

- \frac{1}{u} u + uu = 0 \cdot u

Simplify

- \frac{1}{u} u + uu = 0 \cdot u

Simplify

- \frac{1}{u} u : - 1

- \frac{1}{u} u

Multiply fractions:

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

= - \frac{1 \cdot u}{u}

Cancel the common factor:

u

= - 1

Simplify

uu : u^{2}

uu

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

uu = u^{1 + 1}

= u^{1 + 1}

Add the numbers:

1 + 1 = 2

= u^{2}

Simplify

0 \cdot u : 0

0 \cdot u

Apply rule

0 \cdot a = 0

= 0

- 1 + u^{2} = 0

- 1 + u^{2} = 0

- 1 + u^{2} = 0

Solve

- 1 + u^{2} = 0 : u = 1, u = - 1

- 1 + u^{2} = 0

Move

1

to the right side

- 1 + u^{2} = 0

Add

1

to both sides

- 1 + u^{2} + 1 = 0 + 1

Simplify

u^{2} = 1

u^{2} = 1

For

x^{2} = f (a)

the solutions are

x = f (a), - f (a)

u = 1, u = - 1

1 = 1

1

Apply rule

1 = 1

= 1

- 1 = - 1

- 1

Apply rule

1 = 1

= - 1

u = 1, u = - 1

u = 1, u = - 1

Verify Solutions

Find undefined (singularity) points:

u = 0

Take the denominator(s) of

- \frac{1}{u} + u

and compare to zero

u = 0

The following points are undefined

u = 0

Combine undefined points with solutions:

u = 1, u = - 1

Substitute back

u = sec (x)

sec (x) = 1, sec (x) = - 1

sec (x) = 1, sec (x) = - 1

sec (x) = 1 : x = 2 πn

sec (x) = 1

General solutions for

sec (x) = 1

sec (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} sec (x) 1 \frac{2 3}{3} 22 Undefined - 2 - 2 - \frac{2 3}{3} 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} sec (x) - 1 - \frac{2 3}{3} - 2 - 2 Undefined 22 \frac{2 3}{3}

x = 0 + 2 πn

x = 0 + 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

sec (x) = - 1 : x = π + 2 πn

sec (x) = - 1

General solutions for

sec (x) = - 1

sec (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} sec (x) 1 \frac{2 3}{3} 22 Undefined - 2 - 2 - \frac{2 3}{3} 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} sec (x) - 1 - \frac{2 3}{3} - 2 - 2 Undefined 22 \frac{2 3}{3}

x = π + 2 πn

x = π + 2 πn

Combine all the solutions

x = 2 πn, x = π + 2 πn

Popular Examples

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cos^2(x)-2cos(x)+1=0,0360

cos^{2} (x) - 2 cos (x) + 1 = 0, 0^{\circ} 36 0^{\circ}

sin^4(x)=sin^2(x),0<= x<2pi

sin^{4} (x) = sin^{2} (x), 0 \leq x < 2 π

sin(θ)-0.1cos(θ)=(6.88)/(9.8)

sin (θ) - 0.1 cos (θ) = \frac{6.88}{9.8}

sin(x)=0.3352

sin (x) = 0.3352

Frequently Asked Questions (FAQ)

What is the general solution for (sec(x))/(cos(x))-1=0 ?
The general solution for (sec(x))/(cos(x))-1=0 is x=2pin,x=pi+2pin