What is the general solution for 1/(cos(x))=tan(x) ?

The general solution for 1/(cos(x))=tan(x) is No Solution for x\in\mathbb{R}

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Popular >

1/(cos(x))=tan(x)

Solution

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

Solution

No Solution for x \in R

Solution steps

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

Subtract

tan (x)

from both sides

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

Simplify

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

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

Convert element to fraction:

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

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

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

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

1 - tan (x) cos (x) = 0

Rewrite using trig identities

1 - tan (x) cos (x)

cos (x) tan (x) = sin (x)

cos (x) tan (x)

Use the basic trigonometric identity:

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

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

Simplify

cos (x) \frac{sin ( x )}{cos ( x )} : sin (x)

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

Multiply fractions:

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

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

Cancel the common factor:

cos (x)

= sin (x)

= sin (x)

= 1 - sin (x)

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 for x \in R

Popular Examples

2cos^2(x)-cos(x)-1=0,(0,2pi)

2 cos^{2} (x) - cos (x) - 1 = 0, (0, 2 π)

-3+cos(2x)*2=0

- 3 + cos (2 x) \cdot 2 = 0

1/9+cos^2(x)=1

\frac{1}{9} + cos^{2} (x) = 1

sin(x)=0.9063

sin (x) = 0.9063

11=12+3.5sin((2pi)/(365)t)

11 = 12 + 3.5 sin (\frac{2 π}{365} t)

Frequently Asked Questions (FAQ)

What is the general solution for 1/(cos(x))=tan(x) ?
The general solution for 1/(cos(x))=tan(x) is No Solution for x\in\mathbb{R}