What is the general solution for csc(x)= 1/(sec(x)) ?

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

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Popular >

csc(x)= 1/(sec(x))

Solution

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

Solution

No Solution for x \in R

Solution steps

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

Subtract

\frac{1}{sec ( x )}

from both sides

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

Simplify

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

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

Convert element to fraction:

csc (x) = \frac{csc ( x ) sec ( x )}{sec ( x )}

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

Since the denominators are equal, combine the fractions:

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

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

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

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

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

Express with sin, cos

- 1 + csc (x) sec (x)

Use the basic trigonometric identity:

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

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

Use the basic trigonometric identity:

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

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

Simplify

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

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

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

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

Multiply fractions:

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

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

Multiply the numbers:

1 \cdot 1 = 1

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

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

Convert element to fraction:

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

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

Multiply:

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

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

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

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

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

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

Rewrite using trig identities

1 - cos (x) sin (x)

Use the Double Angle identity:

2 sin (x) cos (x) = sin (2 x)

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

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

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

Move

1

to the right side

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

Subtract

1

from both sides

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

Simplify

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

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

Multiply both sides by

2

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

Multiply both sides by

2

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

Simplify

- sin (2 x) = - 2

- sin (2 x) = - 2

Divide both sides by

- 1

- sin (2 x) = - 2

Divide both sides by

- 1

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

Simplify

sin (2 x) = 2

sin (2 x) = 2

- 1 \leq sin (x) \leq 1

No Solution for x \in R

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Frequently Asked Questions (FAQ)

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