What is the general solution for tan(x)csc(x)=cos(2) ?

The general solution for tan(x)csc(x)=cos(2) is No Solution for x\in\mathbb{R}

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tan(x)csc(x)=cos(2)

tan (x) csc (x) = cos (2)

No Solution for x \in R

Solution steps

tan (x) csc (x) = cos (2)

Subtract

cos (2)

from both sides

tan (x) csc (x) - cos (2) = 0

Express with sin, cos

- cos (2) + csc (x) tan (x)

Use the basic trigonometric identity:

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

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

Use the basic trigonometric identity:

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

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

Simplify

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

- cos (2) + \frac{1}{sin ( x )} \cdot \frac{sin ( x )}{cos ( x )}

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

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

Multiply fractions:

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

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

Cancel the common factor:

sin (x)

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

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

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

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

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

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

1 - cos (2) cos (x) = 0

Move

1

to the right side

1 - cos (2) cos (x) = 0

Subtract

1

from both sides

1 - cos (2) cos (x) - 1 = 0 - 1

Simplify

- cos (2) cos (x) = - 1

- cos (2) cos (x) = - 1

Divide both sides by

- cos (2)

- cos (2) cos (x) = - 1

Divide both sides by

- cos (2)

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

Simplify

cos (x) = \frac{1}{cos ( 2 )}

cos (x) = \frac{1}{cos ( 2 )}

- 1 \leq cos (x) \leq 1

No Solution for x \in R

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0 = 4 + 3 cos (x)

cos^{2} (x) - cos (x) = 0.75

2 sin^{3} (x) + cos^{2} (x) = 1

What is the general solution for tan(x)csc(x)=cos(2) ?
The general solution for tan(x)csc(x)=cos(2) is No Solution for x\in\mathbb{R}