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

The general solution for csc(x)=sec(x) is x= pi/4+pin

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Popular Trigonometry >

csc(x)=sec(x)

Solution

csc (x) = sec (x)

Solution

x = \frac{π}{4} + πn

Degrees

x = 4 5^{\circ} + 18 0^{\circ} n

Solution steps

csc (x) = sec (x)

Subtract

sec (x)

from both sides

csc (x) - sec (x) = 0

Express with sin, cos

csc (x) - sec (x)

Use the basic trigonometric identity:

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

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

Use the basic trigonometric identity:

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

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

Simplify

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

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

Least Common Multiplier of

sin (x), cos (x) : sin (x) cos (x)

sin (x), cos (x)

Lowest Common Multiplier (LCM)

Compute an expression comprised of factors that appear either in

sin (x)

cos (x)

= sin (x) cos (x)

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

sin (x) cos (x)

For

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

multiply the denominator and numerator by

cos (x)

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

For

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

multiply the denominator and numerator by

sin (x)

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

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

Since the denominators are equal, combine the fractions:

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

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

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

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

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

cos (x) - sin (x) = 0

Rewrite using trig identities

cos (x) - sin (x) = 0

Divide both sides by

cos (x), cos (x) \neq = 0

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

Simplify

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

Use the basic trigonometric identity:

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

1 - tan (x) = 0

1 - tan (x) = 0

Move

1

to the right side

1 - tan (x) = 0

Subtract

1

from both sides

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

Simplify

- tan (x) = - 1

- tan (x) = - 1

Divide both sides by

- 1

- tan (x) = - 1

Divide both sides by

- 1

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

Simplify

tan (x) = 1

tan (x) = 1

General solutions for

tan (x) = 1

tan (x)

periodicity table with

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} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

x = \frac{π}{4} + πn

x = \frac{π}{4} + πn

Graph

Popular Examples

sec(x)=3

-sin(x)=cos(x)

sin(3x)+sin(x)=0,0<= x<= 2pi

4sin(θ)+3=1

25cot^2(θ)-1=0

Frequently Asked Questions (FAQ)

What is the general solution for csc(x)=sec(x) ?
The general solution for csc(x)=sec(x) is x= pi/4+pin